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\title{
\textbf{
    IAS/PCMI GSS 2021 Talks
  }
    \\ {\normalsize (Combined)} \\
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2021-07-27 }
\end{flushleft}


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{preface}{%
\section{Preface}\label{preface}}

\begin{quote}
(DZG): These are just some lists and resources I jotted down over the
course of the week, relating to definitions to look up or
topics/conjectures to read more about. I've included them as a vague
``word cloud'', perhaps as a useful way to get some high-level view of
what ideas might show up.
\end{quote}

\begin{quote}
Also, a disclaimer: most of these notes were live-tex'd, and
\emph{almost certainly} contain errors or inaccuracies! Such errors are
most likely my own, due either to hearing and typesetting incorrectly,
or simply misunderstanding. For a \emph{much} more accurate account of
the details for these talks, I'd recommend reading each speaker's own
lecture notes, which I've linked in each relevant section.
\end{quote}

\hypertarget{some-general-resources}{%
\subsection{Some general resources}\label{some-general-resources}}

\begin{itemize}
\tightlist
\item
  \href{https://ncatlab.org/nlab/show/motivic+homotopy+theory}{nLab's
  entry on motivic homotopy theory}

  \begin{itemize}
  \tightlist
  \item
    \href{https://ncatlab.org/nlab/show/Grothendieck+topology}{Covering
    families and Grothendieck topologies}
  \item
    \href{https://www.ias.edu/pcmi/pcmi-2021-gss-lecture-notes-and-problem-sets}{All
    of the actual lecture notes from the authors}
  \end{itemize}
\item
  \href{https://dzackgarza.com/rawnotes/Class_Notes/2021/Summer/PCMI/}{Source
  code and raw files for this project}
\end{itemize}

\hypertarget{definitions}{%
\subsection{Definitions}\label{definitions}}

\begin{itemize}
\tightlist
\item
  \(K{\hbox{-}}\)theory
\item
  Milnor \(K{\hbox{-}}\)theory
\item
  Witt ring
\item
  Period and index
\item
  Symbols
\item
  Brauer group
\item
  Bloch's higher Chow groups
\item
  Mixed characteristic
\item
  Weil Cohomology theory

  \begin{itemize}
  \tightlist
  \item
    Betti cohomology
  \item
    Rigid cohomology
  \item
    Motivic cohomology
  \item
    Étale cohomology
  \item
    Galois cohomology
  \end{itemize}
\item
  Sites

  \begin{itemize}
  \tightlist
  \item
    The smooth site
  \item
    The étale site
  \item
    The Nisnevich site
  \item
    Difference between big and small sites
  \end{itemize}
\item
  Cartier divisors
\item
  Henselian schemes
\item
  Dévissage
\item
  Purity theorems
\item
  Fiber functor
\item
  Transfers
\item
  Torsors
\item
  Discretely valued fields
\item
  Cotangent complex
\item
  Symbols

  \begin{itemize}
  \tightlist
  \item
    Cohomological symbol
  \item
    Tate symbol
  \item
    Galois symbol
  \end{itemize}
\item
  Tate twist
\end{itemize}

\hypertarget{resultsconjectures}{%
\subsection{Results/Conjectures}\label{resultsconjectures}}

\begin{itemize}
\tightlist
\item
  What is \({\mathsf{K}}_n({\mathbb{Z}})\)?
\item
  Kummer-Vandiver conjecture

  \begin{itemize}
  \tightlist
  \item
    For \(h_k\) the class number of the maximum real subfield \(K\) of
    the \(p\)th cyclotomic fields, \(p\notdivides h_k\).
  \end{itemize}
\item
  Milnor conjecture (proved)

  \begin{itemize}
  \tightlist
  \item
    Relates \(H_{ \mathsf{Gal}}\) to to
    \({\mathsf{K}}^{\scriptstyle\mathrm{M}} /2\) and quadratic forms.
  \end{itemize}
\item
  Bloch-Kato conjecture, i.e.~Norm-Residue isomorphism (proved?)

  \begin{itemize}
  \item
    There is an isomorphism induced by the norm-residue map
    \begin{align*}
    h_n:  {\mathsf{K}}^{\scriptstyle\mathrm{M}} _n(F) \to H^n(F; \mu_\ell{ {}^{ \scriptstyle\otimes_{}^{n} }  })
    .\end{align*}
  \item
    Generalized Milnor's conjecture to odd primes.
  \item
    Closely related to Beilinson-Lichtenbaum
  \end{itemize}
\item
  Standard conjectures on algebraic cycles
\item
  Beilinson-Soulé conjecture

  \begin{itemize}
  \tightlist
  \item
    Support in degrees \(d<0\) for \({\mathbb{Z}}_{ \mathrm{mot}} (j)\).
  \item
    Currently unknown
  \end{itemize}
\item
  Gersten conjecture

  \begin{itemize}
  \tightlist
  \item
    Support in degrees \(d\leq 2j\) for
    \({\mathbb{Z}}_{ \mathrm{mot}} (j)\).
  \end{itemize}
\item
  Beilinson-Lichtenbaum conjecture

  \begin{itemize}
  \tightlist
  \item
    There is an isomorphism
    \begin{align*}
    H^p_{ \mathrm{mot}} (X; {\mathbb{Z}}(q)) \xrightarrow{\sim} H^p_\text{ét}(X; {\mathbb{Z}}(1)) && p\leq q
    .\end{align*}
  \end{itemize}
\item
  Vanishing and rigidity conjectures
\item
  Kummer--Vandiver conjecture
\item
  Tate conjecture
\item
  Kato conjecture
\item
  Bass' finite generation conjecture

  \begin{itemize}
  \tightlist
  \item
    Morrow: for \(R\) a regular ring of finite type over
    \({\mathbb{Z}}\), \({\mathsf{K}}_n(R)\) is finitely generated.
  \item
    Wikipedia: the groups \(G_n(A)\) are finitely generated
    \({\mathbb{Z}}{\hbox{-}}\)modules when \(A\) is finitely generated
    as an \({\mathbb{Z}}{\hbox{-}}\)algebra, where \(G_n\) is the
    variant of the \({\mathsf{K}}{\hbox{-}}\)theory construction where
    one takes finitely generated modules instead of projective modules.
  \item
    Generalizes Dirichlet's unit theorem: \({\mathsf{K}}_1(R)\) is
    finitely-generated for \({\mathcal{O}}\) the ring of integers of a
    number field \(K\).
  \end{itemize}
\item
  Parshin's conjecture

  \begin{itemize}
  \tightlist
  \item
    Vanishing of higher \({\mathsf{K}}{\hbox{-}}\)groups for smooth
    varieties over finite fields.
  \end{itemize}
\item
  Quillen-Lichtenbaum conjecture
\item
  Hilbert 90
\end{itemize}

\hypertarget{generic-notes}{%
\subsection{Generic Notes}\label{generic-notes}}

\begin{itemize}
\tightlist
\item
  \({\mathsf{K}}^{\scriptstyle\mathrm{M}}\) has an explicit description
  and is easy to map out of. \(H_{ \mathrm{mot}}\) is difficult in
  general, but usual \({\mathsf{K}}{\hbox{-}}\)theory is filtered by
  \(H_{ \mathrm{mot}}\) pieces.
\item
  Some motivations for \({\mathsf{K}}{\hbox{-}}\)theory:

  \begin{itemize}
  \tightlist
  \item
    Special values of \(L{\hbox{-}}\)functions
  \item
    \({\mathsf{K}}_0\):

    \begin{itemize}
    \tightlist
    \item
      AG: Grothendieck-Riemann-Roch, intersection theory on algebraic
      varieties.
    \item
      NT: \({\operatorname{Pic}}(R)\), the class group
      \({ \operatorname{Cl}} (R)\), class field theory
    \end{itemize}
  \item
    Higher regulators
  \item
    Quadratic reciprocity
  \item
    Embeddings of number fields into \({\mathbb{R}}, {\mathbb{C}}\).
  \item
    Whitehead torsion, used in surgery classification of manifolds and
    the Poincaré conjecture in \(\dim \geq 5\).
  \item
    Sheaf cohomology of a \({\mathsf{K}}{\hbox{-}}\)theory sheaf
    recovers Chow (Bloch's formula)
  \end{itemize}
\item
  The Dennis trace relates \({\mathsf{K}}\to {\operatorname{HH}}\), and
  \({\operatorname{THH}}\) yields an intermediate step.
\end{itemize}

\hypertarget{major-objects}{%
\subsubsection{Major Objects}\label{major-objects}}

\begin{itemize}
\tightlist
\item
  \(H{\mathbb{Z}}\): motivic cohomology.

  \begin{itemize}
  \tightlist
  \item
    Compare to \(H_{ \mathrm{mot}}\)?
  \end{itemize}
\item
  \(H\tilde {\mathbb{Z}}\): generalized motivic cohomology.
\item
  \({\operatorname{CH}}^*\): Chow groups, algebraic codimension \(n\)
  cycles mod rational equivalence.
\item
  \(\tilde{{\operatorname{CH}}}^*\): Chow-Witt groups or oriented Chow,
  formal sums of codimension \(n\) subvarieties with coefficients in
  \({\operatorname{GW}}(k)\) for \(k\) some field.
\item
  \(q_{\operatorname{hyp}}\coloneqq\left\langle{1}\right\rangle + \left\langle{-1}\right\rangle\),
  the hyperbolic form
\item
  \({\operatorname{GW}}(k)\): the Grothendieck-Witt group of \(k\),
  group completion of the semiring of nondegenerate symmetric bilinear
  forms under orthogonal sum.
\item
  \(W(k)\): the Witt group of \(k\),
  \(GW(k)/{\mathbb{Z}}\left\langle{q_{\operatorname{hyp}}}\right\rangle\).
\end{itemize}

\({\mathsf{K}}{\hbox{-}}\)theory:

\begin{itemize}
\tightlist
\item
  \({\mathsf{K}}_*\): \({\mathsf{K}}{\hbox{-}}\)theory.

  \begin{itemize}
  \tightlist
  \item
    \({\mathsf{K}}_0(F) = {\mathbb{Z}}\),
    \({\mathsf{K}}_1(F) = F^{\times}\).
  \item
    \({\mathsf{K}}_0(R) = \mathop{\mathrm{Hom}}_{\mathsf{Top}}(\operatorname{Spec}R, {\mathbb{Z}})\).
    If \(R\) is a domain, \({\mathsf{K}}_0(R) = {\mathbb{Z}}\).
  \item
    \({\mathsf{K}}_0({\mathcal{O}}) \in \operatorname{Ext} ({\mathbb{Z}}, { \operatorname{Cl}} ({\mathcal{O}}))\),
    \({\mathsf{K}}_1({\mathcal{O}}) = {\mathcal{O}}^{\times}\).
  \item
    \({\mathsf{K}}_1(R) = \operatorname{GL}(R)\).
  \item
    For finite fields:
    \begin{align*}
    {\mathsf{K}}_n({\mathbb{F}}_q) = 
    \begin{cases}
    {\mathbb{Z}}& n=0 
    \\
    0 & n \text{ even}
    \\
    {\mathbb{Z}}/\left\langle{q^{ {n+1\over 2} - 1 }}\right\rangle & n \text{ odd}.
    \end{cases}
    \end{align*}
  \end{itemize}
\item
  \({\operatorname{KO}}_*\): Hermitian \({\mathsf{K}}{\hbox{-}}\)theory.
\end{itemize}

\todo[inline]{Include known computations of \({\mathsf{K}}{\hbox{-}}\)theory, $W(k), GW(k)$, etc.}

\hypertarget{danny-krashen-talk-1-monday-july-12}{%
\section{Danny Krashen, Talk 1 (Monday, July
12)}\label{danny-krashen-talk-1-monday-july-12}}

\hypertarget{intro}{%
\subsection{Intro}\label{intro}}

\todo[inline]{Missed first 12m}

\textbf{Abstract}:

\begin{quote}
A fundamental question in field arithmetic is how one can bound, in
various senses, the complexity of algebraic objects such as algebras,
quadratic forms, or cohomology classes. This question is intimately
related to notions of essential dimension, symbol length, and also to
the construction of generic splitting varieties. In these talks, I will
describe some of the principal questions of this sort, and various
methods by which they have been approached.
\end{quote}

\textbf{References}:

\begin{itemize}
\tightlist
\item
  \url{http://dkrashen.github.io/}
\end{itemize}

\begin{remark}

Fix a field \(k_0 \in \mathsf{Field}\), we'll consider extensions
\(k\in\mathsf{Field}_{/ {k_0}}\).

\end{remark}

\hypertarget{galois-cohomology}{%
\subsection{Galois Cohomology}\label{galois-cohomology}}

\begin{definition}[Galois Cohomology]

For \(M\in{\mathsf{G_k}{\hbox{-}}\mathsf{Mod}}\) for \(G_k\) the Galois
group of \(k\in\mathsf{Field}_{/ {k_0}}\), we can take invariants
\(M^{G_k}\). The functor \({-}^{G_k}\) is left-exact, so we define
\begin{align*}
H^*_{ \operatorname{Gal}} (G_k; {-}) \coloneqq{\mathbb{R}}^*({-})^{G_k}
.\end{align*}

\end{definition}

\begin{remark}

Note that the tensor product on \({\mathsf{G_k}{\hbox{-}}\mathsf{Mod}}\)
induces a cup product on \(H^*_{ \mathsf{Gal}}\). An important example
of coefficients is \(M = \mu_\ell^{\otimes m}\), where
\(\mu_\ell^{\otimes 0} \coloneqq{\mathbb{Z}}/n\). It is known that
\(H_{ \mathsf{Gal}} ^*(G_k; \mu^{\otimes 0}) = {\mathbb{Z}}/n\).

We'll define \emph{symbols}
\begin{align*}
(a_1, \cdots, a_n) \coloneqq(a_1) \smile\cdots \smile(a_n) \in H_{ \mathsf{Gal}} ^*(k, \mu_\ell^{\otimes n})
,\end{align*}
which are in fact generators. To remember the \(\ell\), we write
\(({ {a}_1, {a}_2, \cdots, {a}_{n}})_\ell\).

\end{remark}

\begin{remark}

Galois cohomology is a special case of étale cohomology, where for
\(M\in {\mathsf{G_k}{\hbox{-}}\mathsf{Mod}}\),
\begin{align*}
H_{ \mathsf{Gal}} ^n(G_k; M) = H_\text{ét}^n(k; M) = H_\text{ét}^n(\operatorname{Spec}k; M)
.\end{align*}
Étale cohomology works for schemes other than just
\(\operatorname{Spec}k\).

\end{remark}

\hypertarget{milnor-k-theory}{%
\subsection{Milnor K-Theory}\label{milnor-k-theory}}

\begin{definition}[?]

Given \(k\in \mathsf{Field}\), define
\begin{align*}
 {\mathsf{K}}^{\scriptstyle\mathrm{M}} _*(k) \coloneqq\bigoplus _{i=1}^\infty  {\mathsf{K}}^{\scriptstyle\mathrm{M}} _i(k)
\end{align*}
where

\begin{itemize}
\tightlist
\item
  \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _0(k) = {\mathbb{Z}}\)
\item
  \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _1(k) = k^m\), written
  additively as elements \(\left\{{a}\right\}\) on the left-hand side,
  so
  \(\left\{{a}\right\} + \left\{{b}\right\} \coloneqq\left\{{ab}\right\}\).
\item
  It's generated by \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _1(k)\),
  with products written by concatenation:
  \begin{align*}
  \left\{{a_1, \cdots, a_n}\right\} = \left\{{a_1}\right\} \left\{{a_2}\right\} \cdots \left\{{a_n}\right\}
  .\end{align*}
\item
  The only relations are \(\left\{{a, b}\right\} = 0\) when \(a+b=1\),
  motivated by
  \begin{align*}
  (a, b)_\ell = 0 \in H^2_{ \operatorname{Gal}} (k; \mu_\ell^{\otimes 2})
  \iff a+b=1
  .\end{align*}
\item
  There is a map
  \begin{align*}
   {\mathsf{K}}^{\scriptstyle\mathrm{M}} _0(k) &\to H_\text{ét}^*(k; \mu_\ell^{\otimes 0}) \\
  \left\{{a}\right\} &\mapsto (a)
  ,\end{align*}
  and the \textbf{Norm-Residue isomorphism} (formerly the
  \textbf{Bloch-Kato conjecture}) states that this is an isomorphism
  after modding out by \(\ell\), i.e.~
  \begin{align*}
   {\mathsf{K}}^{\scriptstyle\mathrm{M}} _0(k)/\ell \xrightarrow{\sim} H_\text{ét}^*(k; \mu_\ell^{\otimes 0})
  .\end{align*}
\end{itemize}

\end{definition}

\hypertarget{witt-ring}{%
\subsection{Witt Ring}\label{witt-ring}}

\begin{remark}

Assume \(\operatorname{ch}k \neq 2\), so there is a correspondence
between quadratic forms and symmetric bilinear forms given by
polarization:
\begin{align*}
\text{Quadratic forms} &\rightleftharpoons\text{Symmetric bilinear forms} \\
q_b(x) \coloneqq b(x,x) &\mapsfrom b(x, y) \\
q &\mapsto b_q(x,y) \coloneqq{1\over 2}\qty{q(x+y) - q(x) - q(y)}
.\end{align*}
So we'll identify these going forward and write \(q\) for an arbitrary
symmetric bilinear form or a quadratic form. We say \(q\) is
\textbf{nondegenerate} if there is an induced isomorphism:
\begin{align*}
V &\xrightarrow{\sim} V {}^{ \vee }\\
v &\mapsto b_q(v, {-})
.\end{align*}

\begin{quote}
Note that a symmetric bilinear form \(q\) on \(V\) can be regarded as an
element of \(\operatorname{Sym}^2(V {}^{ \vee })\).
\end{quote}

\end{remark}

\begin{definition}[The Witt Ring]

Let \({\mathsf{QuadForm}}_{/ {k}}\) be the category of \textbf{quadratic
spaces}: pairs \((V, q)\) with \(V\in \mathsf{VectSp}_{/ {k}}\) a
\(k{\hbox{-}}\)vector space and
\(q\in \operatorname{Sym}^2(V {}^{ \vee })\) representing a quadratic
form on \(V\). The \textbf{Witt ring} is generated as a group by
isomorphism representing a quadratic form on \(V\).
\begin{align*}
W(k) = \frac{{\mathbb{Z}}\left\langle{\left\{{ [(V, q)] \in {\mathsf{QuadForm}}_{/ {k}}  }\right\} }\right\rangle}{
\left\langle{
q_{\operatorname{hyp}}, (q_1 + q_2) - (q_1 \perp q_2)
}\right\rangle 
}
\in {\mathsf{Ab}}{\mathsf{Grp}}
.\end{align*}
where the \textbf{hyperbolic form} is defined as
\(q_{\operatorname{hyp}}(x, y) = xy\). The ring structure is given by
the tensor product (a.k.a. Kronecker product of forms).

\end{definition}

\begin{remark}

Noting that Galois cohomology lives mod \(\ell\) for various \(\ell\),
here \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _0(k)\) lives over
\({\mathbb{Z}}\). So Milnor K-theory relates all of the various mod
\(\ell\) Galois cohomologies together.

\end{remark}

\begin{definition}[Fundamental ideals and Pfister Forms]

The \textbf{fundamental ideal} \(I(k) {~\trianglelefteq~}W(k)\) is the
ideal of even dimensional forms, and set \(I^n(k) \coloneqq(I(k))^n\).
There is a map
\begin{align*}
 {\mathsf{K}}^{\scriptstyle\mathrm{M}} _n(k) &\to I^n(k) / I^{n+1}(k) \\
\left\{{{ {a}_1, {a}_2, \cdots, {a}_{n}}}\right\} &\mapsto \left\langle{\left\langle{ { {a}_1, {a}_2, \cdots, {a}_{n}} }\right\rangle}\right\rangle
,\end{align*}
which follows from Gram-Schmidt: any form can be diagonalized
\(q \cong \sum a_i x_i^2\), which we can write as
\(\left\langle{{ {a}_1, {a}_2, \cdots, {a}_{n}}}\right\rangle\). We can
define the \textbf{\(n{\hbox{-}}\)fold Pfister forms}
\begin{align*}
\left\langle{\left\langle{ a }\right\rangle}\right\rangle &\coloneqq\left\langle{\left\langle{1, -a}\right\rangle}\right\rangle \\
\left\langle{\left\langle{{ {a}_1, {a}_2, \cdots, {a}_{n}}}\right\rangle}\right\rangle &\coloneqq\prod_{i=1}^n \left\langle{\left\langle{ a_i }\right\rangle}\right\rangle
.\end{align*}

\end{definition}

\begin{remark}

The \textbf{Milnor conjecture} (proved by Voevodsky et al) states that
the above map is an isomorphism after modding out by 2, so
\begin{align*}
 {\mathsf{K}}^{\scriptstyle\mathrm{M}} _n(k)/2 \xrightarrow{\sim} I^n(k) / I^{n+1}(k)
.\end{align*}
Moreover, the LHS is isomorphic to \(H^n(k, \mu_2)\). There are
interesting maps going the other way
\begin{align*}
I^n(k) \to I^n(k) / I^{n+1}(k) \xrightarrow{\sim} H^n(k, \mu_2)
\end{align*}

Upshot: this is surjective -- any mod \(2\) cohomology class comes from
a quadratic form, and thus we can reason about cohomology by reasoning
about quadratic forms.

\end{remark}

\hypertarget{motivic-cohomology}{%
\subsection{Motivic Cohomology}\label{motivic-cohomology}}

\begin{remark}

\textbf{Motivic cohomology} relates the various mod \(\ell\)
cohomologies together much like
\({\mathsf{K}}^{\scriptstyle\mathrm{M}} _*\), but additionally relates
different twists. In particular, it relates various
\(H^i_\text{ét}(k; \mu_\ell^{\otimes j})\), where Milnor K-theory
interprets this ``diagonally'' when \(i=j\). This works by constructing
\textbf{motivic complexes}
\begin{align*}
{\mathbb{Z}}(m) \in \mathsf{Ch}( \underset{ \mathsf{pre} } {\mathsf{Sh} }{\mathsf{sm}}{\mathsf{Sch}}_{/ {k}} )
,\end{align*}
which are complexes of presheaves on smooth \(k{\hbox{-}}\)schemes,
usually considered in the Zariski, étale, or Nisnevich topologies.

\end{remark}

\begin{remark}

\textbf{Zariski hypercohomology} is defined as
\begin{align*}
{\mathbb{H}}^n(X; {\mathbb{Z}}(m)) = H^{n, m}(X; {\mathbb{Z}}) = H_{ \mathrm{mot}} ^n(X; {\mathbb{Z}}(m)) && \text{for } X\coloneqq\operatorname{Spec}k
.\end{align*}
These relate to Galois cohomology in the following ways:

\begin{itemize}
\tightlist
\item
  There is a quasi-isomorphism
  \(\mu_\ell^{\otimes m} \xrightarrow{\sim_W} {\mathbb{Z}}/\ell(n)\) in
  the étale topology.
\item
  There is an isomorphism
  \(H^n_{\mathrm{zar}}(k, {\mathbb{Z}}(n)) \xrightarrow{\sim} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _n(k)\).
\item
  Bloch-Kato identifies
  \(H_{\mathrm{zar}}^*(X; {\mathbb{Z}}/\ell(n)) \xrightarrow{\sim} H_\text{ét}^n(X; {\mathbb{Z}}/\ell(n))\).
\end{itemize}

\end{remark}

\hypertarget{dimension}{%
\subsection{Dimension}\label{dimension}}

\begin{remark}

There are a number of competing notions for the ``dimension'' of a
field.

\end{remark}

\begin{definition}[Dimension of a field]

If \(k\) is finitely generated over either a prime field or an
algebraically closed field, we say
\begin{align*}
\dim(k) 
=
\begin{cases}
[k: k_0]_{\mathrm{tr}}& k_0 = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu_0  
\\
[k: k_0]_{\mathrm{tr}}+1 & k_0 \text{ finite} \\
[k: k_0]_{\mathrm{tr}}+ 2 & k_0 = {\mathbb{Q}}.
\end{cases}
\end{align*}

\end{definition}

\begin{definition}[Cohomological dimension]

We define its \textbf{cohomological dimension}
\(\operatorname{cohdim}(k)\), which is at most \(n\) if
\(H^n(G_k; M) = 0\) for all \(m>n\) and \(M\) torsion,
\begin{align*}
\operatorname{cohdim}(k) \coloneqq\min \left\{{n {~\mathrel{\Big|}~}\operatorname{cohdim}(k) \leq n}\right\}
.\end{align*}
Equivalently, \(\operatorname{cohdim}(k) = n \iff\) there exists a
torsion \(M\) with \(H^n(G_k; M) \neq 0\) and \(H^m(G_k; M) = 0\) for
all \(m>n\).

\end{definition}

\begin{remark}

\(\operatorname{cohdim}(k) = \dim(k)\) if \(k\) is finitely generated or
a finite extension of
\(k_0 = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu_0\),
or if \(k\) is finitely generated over \({\mathbb{Q}}\) and has no real
orderings. So if \(k\) has orderings,
\(\operatorname{cohdim}(k) = \infty\).

\end{remark}

\begin{definition}[Diophantine Dimension]

We say \(k\) is \(C_n\) if for \(d>0\) and \(m>d^n\), then every
homogeneous polynomials of degree \(d\) in \(m\) variables has a
nontrivial root.
\begin{align*}
\operatorname{ddim}(k) \coloneqq\min\left\{{n{~\mathrel{\Big|}~}k \text{ is } C_n}\right\}
.\end{align*}

\end{definition}

\begin{example}[?]

If \(k\) is finitely generated or finite over
\(k_0 = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu_0\),
then
\begin{align*}
\operatorname{ddim}(k) = \dim(k) = \operatorname{cohdim}(k)
.\end{align*}

\end{example}

\begin{definition}[$T_n\dash$rank]

We say \(k\) is \(T_n\) if for every
\({ {d}_1, {d}_2, \cdots, {d}_{r}} > 0\) and every system of polynomial
equations \(f_1 = \cdots = f_r = 0\) with \(\deg f_i = d_i\) in \(m\)
variables, with \(m > \sum d_i^n\). Then the
\textbf{\(T_n{\hbox{-}}\)rank} is defined as
\begin{align*}
T_n{\hbox{-}}\operatorname{rank}(k) \coloneqq\min\left\{{n {~\mathrel{\Big|}~}k \text{ is } T_n}\right\}
.\end{align*}

\end{definition}

\begin{question}

Note that \(T_n\implies C_n\), so
\(T_n{\hbox{-}}\operatorname{rank}(k) \geq \operatorname{ddim}(k)\),
when are they equal? This is likely unknown.

\end{question}

\begin{remark}

There is a famous example of a field \(k\) with
\(\operatorname{cohdim}(k)=1\) but \(\operatorname{ddim}(k) = \infty\).

\end{remark}

\begin{question}

Is it true that
\(\operatorname{ddim}(k) \geq \operatorname{cohdim}(k)\)? Serre showed
that this holds when \(\operatorname{cohdim}\) is replaced by
\(\operatorname{cohdim}_2\), the 2-primary part -- does this hold for
all \(p\)? These are both open.

Why would one expect this to be true?

\end{question}

\begin{remark}

A recent result: \(\operatorname{cohdim}_p\) grows at most linearly in
\(\operatorname{ddim}\), with slope not 1 but rather
\(\approx \log_2 p\). These questions say that if an equation has enough
variables then there is a solution, but why should this be reflected in
cohomology? To show this bound, one would want to show that given some
\(\alpha \in H^*(k)\), there exists a polynomial \(f_\alpha\) where if
\(f_\alpha\) has a root and \(\alpha = 0\) in homology. In special
cases, we were able to come up with such polynomials. When \(\alpha\) is
a symbol, this is closely related to \emph{norm varieties} which have a
point iff \(\alpha\) is split. One might optimistically hope these are
described as hypersurfaces, from which answers to the above would
follow, but they turn out to not have such a concrete realization.

\end{remark}

\hypertarget{structural-problems-in-galois-cohomology}{%
\subsection{Structural Problems in Galois
Cohomology}\label{structural-problems-in-galois-cohomology}}

\begin{remark}

Here we'll describe the problems we need help with! Perhaps insight from
motivic cohomology will lend insight to them. We'll write
\(H^i(k) \coloneqq H^i(k; \mu_\ell^{\otimes j})\).

\end{remark}

\hypertarget{period-index-problems}{%
\subsubsection{Period-Index Problems}\label{period-index-problems}}

\begin{definition}[An extension splitting a cohomology class]

If \(\alpha \in H^i(k)\), we say \(L_{/ {k}}\) \textbf{splits}
\(\alpha\) if
\begin{align*}
{ \left.{{\alpha}} \right|_{{L}} } = 0\in H^i(L)
.\end{align*}

\end{definition}

\begin{definition}[?]

We define the \textbf{index}
\begin{align*}
\mathop{\mathrm{ind}}\alpha \coloneqq\gcd\left\{{[L:k] {~\mathrel{\Big|}~}L_{/ {k}}  \text{ finite and splits } \alpha}\right\}
.\end{align*}
and the \textbf{period} of \(\alpha\) as its (group-theoretic) order
\(H^i(k)\). Note that \(\mathop{\mathrm{per}}\alpha \leq \ell\).

\end{definition}

\begin{remark}

One can show that
\(\mathop{\mathrm{per}}\alpha \divides \mathop{\mathrm{ind}}\alpha\),
and
\(\mathop{\mathrm{ind}}\alpha \divides \qty{\mathop{\mathrm{per}}\alpha}^m\)
for some \(m\).

\end{remark}

\begin{question}

For a fixed \(k\) and \(i, j, \ell\), which is the minimum \(m\) such
that
\begin{align*}
\mathop{\mathrm{ind}}\alpha \divides \qty{\mathop{\mathrm{per}}\alpha}^m
?\end{align*}
Alternatively, what is the minimum \(m\) such that
\(\mathop{\mathrm{ind}}\alpha \divides \ell^m\)?

\end{question}

\begin{conjecture}

If \(\operatorname{ddim}(k) = n\) (or \(\dim(k) = n\) since \(k\) is
finitely generated) with \(\alpha \in H^2(k, \mu_\ell)\), then
\begin{align*}
\mathop{\mathrm{ind}}\alpha \divides \qty{\mathop{\mathrm{per}}\alpha}^{n-1}
.\end{align*}

\end{conjecture}

\begin{remark}

Even in this case, no known bound is known for \(k = {\mathbb{Q}}(t)\),
for any choice of \(\ell\). How complicated can the cohomology class be?
The rough idea is that for \(H^i(k)\) with \(i\) near \(\dim k\), this
should have a small index and if \(i=\dim k\) then
\(\mathop{\mathrm{per}}k = \mathop{\mathrm{ind}}k\).

\end{remark}

\begin{remark}

We know \(\mathop{\mathrm{per}}= \mathop{\mathrm{ind}}\) for any number
field for classes in \(H^2(\operatorname{Spec}k; \mu_N)\), with or
without roots.

\end{remark}

\hypertarget{symbol-length-problem}{%
\subsubsection{Symbol Length Problem}\label{symbol-length-problem}}

\begin{remark}

We know \(H^n(k, \mu_\ell^{\otimes n})\) is generated by symbols
\(({ {a}_1, {a}_2, \cdots, {a}_{n}})\). We can use symbol length to
measure complexity, leading to the following:

\end{remark}

\begin{question}

Given \(k, n\), what is the minimal number \(m\) such that every
\(\alpha\in H^n(k)\) is a sum of no more than \(m\) symbols. I.e. how
easy is it to write \(\alpha\)?

\end{question}

\begin{remark}

We'd like a bound in terms of \(\operatorname{ddim}(k)\) and
\(\dim(k)\). One can construct fields needing arbitrarily long symbols,
but perhaps for finite dimensional fields, one feels there should be a
bound. Danny feels that there may not be such a bound once \(n\geq 4\).

\end{remark}

\begin{remark}

What's known: for number fields (or global fields, i.e.~a reasonable
notion of dimension with \(\dim k = 2\)) which lie over finitely
generated or prime fields and have a primitive \(\ell\)th root of unity,
we know every class in \(H^2\) can be written as exactly one symbol.

\end{remark}

\begin{remark}

A result of Malgri (?): assuming we have roots of unity, if
\(\ell = p^t\), then for \(H^2\) one needs at most
\(t(p^{\operatorname{ddim}(k)-1}-1)\) symbols. If
\(\operatorname{ddim}(k)< \infty\) this yields a bound, and
conjecturally this shouldn't depend on ???

For higher degree cohomology, we know almost nothing except for special
cases of \(H^4\) for ``3-dimensional'' \(p{\hbox{-}}\)adic curves.

\end{remark}

\begin{remark}

If one can bound the symbol length, one can uniformly write down a
generic element in cohomology as a sum of at most \(n\) symbols. The
inability to be able to write down a general form of a cohomology class
for a given field is what makes this difficult -- they have
``complexity'' that isn't necessarily bounded in a known way.

\end{remark}

\hypertarget{danny-krashen-talk-2-tuesday-july-13}{%
\section{Danny Krashen, Talk 2 (Tuesday, July
13)}\label{danny-krashen-talk-2-tuesday-july-13}}

\hypertarget{setup}{%
\subsection{Setup}\label{setup}}

\begin{remark}

Fix a \(k_0\in\mathsf{Field}\).

\textbf{Outline}

\begin{itemize}
\item
  Arithmetic problems: consider ``complexity'' of cohomology or
  algebraic structures (Witt group, symbol length, index of classes).

  \begin{itemize}
  \tightlist
  \item
    Examples were \(\operatorname{ddim}, \operatorname{cohdim}\), the
    period-index problem, the period-symbol length problem, which we saw
    last time.
  \end{itemize}
\item
  Algebraic structure problems: describe (algebraic) structural features
  of the class of all field extensions
  \(k \in\mathsf{Field}_{/ {k_0}}\).
\end{itemize}

Today we'll describe a way to connect these using a notion of
\emph{essential dimension}. Computing this is difficult in general, but
finding lower/upper bounds can be tractable. We'll get upper bounds from
\emph{canonical dimensions}, and lower bounds from cohomological
invariants.

\end{remark}

\hypertarget{symbol-length}{%
\subsection{Symbol Length}\label{symbol-length}}

\begin{remark}

For a particularly concrete problem, consider
\begin{align*}
\alpha\in H^2(k; \mu_\ell) \subseteq H^2(k; {\mathbb{G}}_m)[\ell] \coloneqq\mathop{\mathrm{Br}}(k)[\ell]
,\end{align*}
i.e.~this is a subgroup of the \(\ell{\hbox{-}}\)torsion of the
\textbf{Brauer group}. Suppose we know
\begin{align*}
\mathop{\mathrm{ind}}\alpha \coloneqq\gcd\left\{{[L:k] {~\mathrel{\Big|}~}\alpha_L = 0}\right\} = \min\left\{{[L:K] {~\mathrel{\Big|}~}\alpha_L = 0}\right\}
,\end{align*}
where the last equality holds in the special case of
\(\mathop{\mathrm{Br}}(k)\). If \(k\) contains a primitive \(\ell\)th
root of unity, we can identify
\(\mu_\ell = {\mathbb{Z}}/\ell = \mu_\ell^{\otimes 2}\), and thus
identify
\begin{align*}
H^2(k; \mu_\ell) = H^2(k; \mu_\ell^{\otimes 2}) = K_2^M(k)/\ell
.\end{align*}
So we can write \(\alpha = \alpha_1 + \cdots + \alpha_r\) as a sum of
symbols with \(\alpha_i = (b_i, c_i)_\ell\) with
\(b_i, c_i\in k^{\times}\).

\end{remark}

\begin{question}

How big does \(n\) have to be?

\end{question}

\begin{remark}

It follows from ``the literature'' (after stringing several results
together) that there almost exists an absolute bounds depending only on
\(\ell, n\) but not \(k\). However, we do not know what this bound
actually is. There are some known cases:

\begin{itemize}
\tightlist
\item
  \(\ell = n = 2, 3\): \(r\leq 1\), so only one symbol is needed.
\item
  \(\ell = n = 4\): probably \(r\leq 4\).
\item
  \(\ell = 2, n=4\): \(r\leq 2\), a classical results on central simple
  algebras.
\item
  \(\ell = 2, n=8: r\leq 4\)
\end{itemize}

\end{remark}

\begin{remark}

It turns out that if \(k\) contains a field \(k_0\) with
\(\operatorname{ddim}k_0 < \infty\), one can produce an explicit bound.
Given some \(\alpha \in H^2(k; \mu_\ell)\) we can find some
\(k_0 \subseteq L \subseteq k\) with \(L\) finitely generated over
\(k_0\) and \([L:k_0]_{\mathrm{tr}}\) depending only on the period
\(\ell\) and index \(n\), such that
\(\alpha \in \operatorname{im}\qty{H^2(L;\mu) \to H^2(k; \mu)}\).

\begin{slogan}

Central simple algebras of a given period and index have finite
essential dimension.

\end{slogan}

An important property is that
\begin{align*}
\operatorname{ddim}L \leq \operatorname{ddim}k_0 + [L: k_0]_{\mathrm{tr}}
.\end{align*}
Recall that we can bound the symbol length in \(H^2(k; \mu_\ell)\) in
terms of \(\operatorname{ddim}L\). The idea is that we can bound the
transcendence degree in terms of \(\ell, n\). This bound can be made
very explicit, although it's not tight: for \(\ell = 2, n=8\), it's
\(2^{17 + \operatorname{ddim}k_0} -1\). This is an improvement over
\(k_0 = {\mathbb{Q}}\) though, where it's known there's a bound but it
can't be written down. The lower bound is \emph{very} low: it is hard to
show a symbol can not be written with very few symbols.

\end{remark}

\hypertarget{pfister-forms}{%
\subsection{Pfister Forms}\label{pfister-forms}}

\begin{remark}

Recall \(W(k)\), whose elements are isometry classes of nondegenerate
quadratic forms with addition given by perpendicular sum and the
Kronecker product. There is a hyperbolic form \(xy\), or \(x^2-y^2\) in
\(\operatorname{ch}k \neq 2\), which we can write as
\(\left\langle{1, -1}\right\rangle\), and a fundamental ideal of
even-dimensional forms
\(\left\langle{1, -a}\right\rangle = \left\langle{\left\langle{ a }\right\rangle}\right\rangle\).
We write
\begin{align*}
\left\langle{\left\langle{ { {a}_1, {a}_2, \cdots, {a}_{n}} }\right\rangle}\right\rangle \coloneqq\left\langle{\left\langle{ a_1 }\right\rangle}\right\rangle \left\langle{\left\langle{ a_2 }\right\rangle}\right\rangle\cdots \left\langle{\left\langle{ a_n }\right\rangle}\right\rangle\in I^n(k)
,\end{align*}
which in fact generate \(I^n(k)\).

\end{remark}

\begin{question}

Given \(q\in I^n(k)\) of dimension \(d\), we know we can write
\(q \sim q_1 \perp \cdots \perp q_r\) where \(q_i\) are
\(n{\hbox{-}}\)fold Pfister forms. How many are needed? Is this number
even bounded?

\end{question}

\begin{theorem}[(Vishik)]

If \(d < 2^n + 2^{n-1}\) then \(r\) is bounded by some small number.

\end{theorem}

\begin{remark}

For \(d\geq 2^n + 2^{n-1}\), it's not so clear, although it is bounded
when \(n\geq 3\). Why is \(n\leq 3\) easy and \(n\geq 4\) hard?

\end{remark}

\begin{remark}

Consider the following objects:

\begin{itemize}
\tightlist
\item
  \(H^2(k; \mu)\)
\item
  \(\mathop{\mathrm{Br}}(k)\)
\item
  \(W(k)\)
\item
  \(I^n(k)\)
\item
  \(q\in I^n(k)\) with \(\dim q = d\)
\end{itemize}

These can all be viewed as functors
\(\mathsf{Field}{_{/ {k_0}} }\to {\mathsf{Set}}\) taking field
extensions to sets.

\end{remark}

\begin{definition}[Essential dimension of a functor]

Given a functor \(f\) and \(\alpha \in F(k)\), define
\begin{align*}
\operatorname{essdim}(\alpha) &= \min\left\{{[L:k_0]_{\mathrm{tr}}{~\mathrel{\Big|}~}\alpha\in \operatorname{im}(F(L) \to F(k)) }\right\}\\
\operatorname{essdim}(F) &= \min\left\{{\operatorname{essdim}(\alpha) {~\mathrel{\Big|}~}\alpha \in F(k) \,\,\forall k_{/ {k_0}} }\right\}
.\end{align*}

\end{definition}

\begin{definition}[Versal]

Given a functor \(F: {\mathsf{Alg}}_{_{/ {k_0}} }\to {\mathsf{Set}}\),
we say \(\alpha \in F(R)\) is \textbf{versal} if for every
\(\beta \in F(K)\), for any \(k_{/ {k_0}}\), there exists a morphism
\(R\to k\) such that \(\beta\) is the image of \(\alpha\) under
\(F(R)\to F(k)\).

\includegraphics{figures/2021-07-13_13-54-39.png}

\end{definition}

\begin{observation}

If there exists a versal \(\alpha \in F(R)\) then
\(\operatorname{krulldim}R \geq \operatorname{essdim}(F)\), so the
essential dimension is bounded above by the Krull dimension.

\end{observation}

\begin{example}[?]

Let \(F(k)\) be the set of quadratic forms of dimension \(n\) over
\(k\), then \(\operatorname{essdim}F = n\). Every such \(q\) can be
diagonalized to yields
\(q \simeq \left\langle{{ {a}_1, {a}_2, \cdots, {a}_{n}}}\right\rangle\)
which is defined over
\(L \coloneqq k_0({ {a}_1, {a}_2, \cdots, {a}_{n}})\). Alternatively,
\begin{align*}
q = \left\langle{{ {x}_1, {x}_2, \cdots, {x}_{n}}}\right\rangle / k_0[x_1^{\pm 1}, x_2^{\pm 1}, \cdots, x_n^{\pm 1} ]
\end{align*}
is versal. Thus every such quadratic form comes from ``specializing''.

Considering now the fundamental ideals, the Milnor conjectures yield an
isomorphism \(I^n/I^{n+1} \cong H^n(k; \mu_2)\), so there is a SES
\begin{align*}
1 \to I^{n+1} \to I^n \xrightarrow{e_n} H^n(k; \mu_2) \to 1
.\end{align*}
Thus a quadratic form \(q\) of dimension \(d\) in \(I^{n+1}\) is
equivalent to \(q\in I^{n}\) such that \(e_n(q) = 0\).

\end{example}

\hypertarget{canonical-dimension}{%
\subsection{Canonical Dimension}\label{canonical-dimension}}

\begin{definition}[Canonical Dimension]

This is a generalization of \(\operatorname{essdim}\). Letting
\(k_{/ {k_0}}\), suppose
\(F: \mathsf{Field}_{/ {k}} \to {\mathsf{Set}}_+\) is a functor now from
extensions of \(k\) (not \(k_0\)) into pointed sets. For
\(\alpha\in F(k)\), define a new functor
\begin{align*}
\check{F}_\alpha(L) \coloneqq
\begin{cases}
\emptyset & \alpha_L \neq {\operatorname{pt}}
\\
\left\{{{\operatorname{pt}}}\right\}  & \alpha_L = {\operatorname{pt}},
\end{cases}
\end{align*}
and define the \textbf{canonical dimension}
\begin{align*}
\operatorname{candim}(\alpha) = \operatorname{essdim}(\check{F}(\alpha))
.\end{align*}

\end{definition}

\begin{remark}

This measures how many parameters are needed to trivialize/split
\(\alpha\). To have \(\operatorname{candim}(\alpha) \leq r\) means that
if \(\alpha = {\operatorname{pt}}\) means the following: if
\(\alpha_L = {\operatorname{pt}}\) then there exists an \(E\) with
\(k \subseteq E \subseteq L\) with \([E:k]_{\mathrm{tr}}\leq r\) such
that \(\alpha_E = {\operatorname{pt}}\).

\end{remark}

\begin{definition}[Generic splitting scheme]

Given \(F\) as above and \(\alpha\in F(k)\), we say an
\(X\in {\mathsf{Sch}}_{/k}\) is a \textbf{generic splitting scheme} for
\(\alpha\) if
\begin{align*}
\alpha_L = 0 \iff X(L) \neq \emptyset
.\end{align*}

\end{definition}

\begin{remark}

So this encodes triviality of \(\alpha\) into polynomial equations.

\end{remark}

\begin{example}[?]

If \(X\) is a generic splitting scheme for \(\alpha\) finite type over
\(L\) implies \(\operatorname{candim}(\alpha) \leq \dim X\).

\end{example}

\hypertarget{splitting-schemes}{%
\subsection{Splitting Schemes}\label{splitting-schemes}}

\begin{question}

Does there exists a finite type generic splitting scheme for cohomology
classes in \(H^i(k; \mu_\ell^{\otimes j})\)?

\end{question}

\begin{remark}

We do know this in special cases:

\begin{itemize}
\tightlist
\item
  \(i=1\): Yes, these are etale algebras, so finite schemes over \(k\).
\item
  \(i=2\): Yes, Danny shows these exist for all twists.

  \begin{itemize}
  \tightlist
  \item
    \(j=1\): Classical, these are Severi-Brauer varieties.
  \end{itemize}
\item
  For symbols, \(i=3,j=2,\ell\) a prime: see Merkurjev-Suslin
\item
  For symbols, \(i=4, j=3, \ell =3\): see Albert algebras
\item
  For symbols, \(\ell\) prime: this can be done up to prime-to-\(\ell\)
  extensions, see \href{https://arxiv.org/abs/math/0304208}{Rost's
  ``Norm Varieties''}. Related to Bloch-Kato conjecture.
\item
  For symbols, \(\ell=2\): see Pfister quadrics.
\end{itemize}

\end{remark}

\begin{remark}

Upshot: if there exists generic splitting schemes for classes in
\(H^i(k; \mu_2)\) for \(i\geq 3\), one could bound Pfister numbers and
thus \(\operatorname{essdim}\). Write \({\mathcal{I}}_d^n(k)\) to be the
set of quadratic forms of dimension \(d\) in \(I^n\), then
\(\operatorname{essdim}({\mathcal{I}}_d^n) < \infty\) would imply that
if \(q\in {\mathcal{I}}_d^n(k)\) for \(k\supseteq k_0\) then \(q\) would
be defined over some \(L_{/ {k_0}}\) with
\([L: k_0]_{\mathrm{tr}}< \infty\).

If we knew that \(\operatorname{ddim}k_0 < \infty\), e.g.~if \(k_0\)
contains a finite field, this yields a bound on \(\operatorname{ddim}L\)
and thus on \(\operatorname{cohdim}L\). If there is a versal element in
\(\alpha\in {\mathcal{I}}^n_d\), then \(\alpha\) needs some finite
number \(m\) of Pfister forms to be written. Everything else is a
specialization of \(\alpha\), so the length \(m\) will almost give an
upper bound.

\end{remark}

\begin{warnings}

This may seem like a correct argument, but it is not! A problem arises
where you may have denominators -- specialization can get worse, but
only a finite number of times, which is how the actual argument goes.

\end{warnings}

\begin{remark}

If you knew the essential dimensions were finite with some given bound,
and some general period-index conjecture were known, these would give
bounds on symbol length in \(H^i(L; \mu_2)\). There's an argument
pushing things into higher powers of the fundamental ideal, thus higher
degree cohomology, which disappear at some point and yield a bound.
Motives enter the picture in terms of the tools used to attack these
problems.

\end{remark}

\hypertarget{phillippe-gille-talk-1-monday-july-12}{%
\section{Phillippe Gille, Talk 1 (Monday, July
12)}\label{phillippe-gille-talk-1-monday-july-12}}

\hypertarget{intro-1}{%
\subsection{Intro}\label{intro-1}}

\textbf{Abstract}:

\begin{quote}
We shall present the theory of G-torsors (or G-bundles) in algebraic
geometry which includes for example vector bundles and quadratic bundles
(Grothendieck-Serre, 1958). We focus on the case of an affine smooth
connected curve firstly over an algebraically closed field k; we shall
show that G-torsors are trivial for a semisimple k-group G. Secondly we
will consider the case of a perfect field and discuss the important case
of the affine line (Raghunathan-Ramanathan, 1984). This will be an
opportunity to deal with étale cohomology and patching techniques.
\end{quote}

\textbf{References:}

\begin{itemize}
\item
  \url{http://math.univ-lyon1.fr/homes-www/gille/}
\item
  \url{http://math.univ-lyon1.fr/homes-www/gille/prenotes/gille_pcmi_part1.pdf}
\item
  V. Chernousov, P. Gille, A. Pianzola, Three-point Lie algebras and
  Grothendieck's dessins d'enfants. Math. Res. Lett. 23 (2016), 81--104.
\item
  J.S. Milne, Lectures on etale cohomology,
  \url{https://www.jmilne.org/math/CourseNotes/LEC.pdf}
\item
  M. S. Raghunathan, A. Ramanathan, Principal bundles on the affine
  line. Proc. Indian Acad. Sci. Math. Sci. 93 (1984), 137--145.
\end{itemize}

\textbf{Notation}:

\begin{itemize}
\tightlist
\item
  \({\mathsf{Bun}}_{G}\) is the category of \(G{\hbox{-}}\)bundles

  \begin{itemize}
  \tightlist
  \item
    \({\mathsf{Bun}}_{\operatorname{GL}_r({\mathbb{R}})}\) is the
    category of real vector bundles of rank \(r\).
  \item
    \({ {\mathsf{Bun}}_{\operatorname{GL}_r} }\) is the category of
    vector bundles of rank \(r\) over an unspecified field.
  \end{itemize}
\item
  \(\mathsf{C}_{/ {R}}\) is an overcategory/slice category of objects
  over a fixed object \(R\).
\item
  \({\mathsf{Sch}}_{/ {S}}\) is the category of schemes over a fixed
  scheme \(S\).

  \begin{itemize}
  \tightlist
  \item
    \({\mathsf{Aff}}{\mathsf{Sch}}\) is the category of affine schemes.
  \item
    \({\mathsf{Sch}}_{/ {R}}\) are the schemes over
    \(\operatorname{Spec}R\).
  \end{itemize}
\item
  \({\mathsf{sm}}\) denotes subcategories of smooth objects
\item
  \(R \left[ { \scriptstyle {\mathfrak{p}}^{-1}} \right]\) is the
  localization of \(R\) at \({\mathfrak{p}}\).
\item
  \({\mathsf{cAlg}}\) denotes commutative algebras.
\item
  An \(R{\hbox{-}}\)functor is an object in
  \({\mathsf{Fun}}({\mathsf{Alg}}_{/ {R}} , {\mathsf{Set}})\), which
  \({\mathsf{Aff}}{\mathsf{Sch}}\) embeds into as representable
  \(R{\hbox{-}}\)functors.

  \begin{itemize}
  \tightlist
  \item
    Todo: so any presheaf on \(R{\hbox{-}}\)algebras..?
  \end{itemize}
\end{itemize}

\hypertarget{serre-swan-and-vector-group-schemes}{%
\subsection{Serre-Swan and Vector Group
Schemes}\label{serre-swan-and-vector-group-schemes}}

\begin{remark}

We'll be looking at ways to go from the world of differential geometry
to algebraic geometry. Notably, in differential geometry we have notions
of

\begin{itemize}
\tightlist
\item
  Vector bundles
\item
  Principal \(G{\hbox{-}}\)bundles
\item
  Principal homogeneous spaces
\end{itemize}

Serre-Grothendieck gave algebro-geometric analogs of these in 1958,
extending these notions to the setting of \(G{\hbox{-}}\)bundles over a
scheme using the étale topology. Today we'll work over rings, or
equivalently affine schemes, since most questions will be local. We'll
in fact restrict to smooth affine curves over a field.

\end{remark}

\begin{remark}

Recall the Serre-Swan correspondence between projective finite modules
of finite rank and smooth vector bundles: for \(X\) paracompact, there
is an equivalence of categories induced by taking global sections:
\begin{align*}
\underset{\operatorname{GL}_n({\mathbb{R}})}{ {\mathsf{sm}}{\mathsf{Bun}}_{/ {X}}  } &\rightleftharpoons\mathsf{C^\infty(X; {\mathbb{R}})}{\hbox{-}}\mathsf{Mod}^{{\mathrm{fg}}, \mathop{\mathrm{proj}}} \\
({\mathcal{E}}\to X) &\mapsto \Gamma(X; {\mathcal{E}})
.\end{align*}
We'll upgrade this to a statement about affine schemes.

\end{remark}

\begin{definition}[Vector Group Scheme]

Let \(R \in \mathsf{CRing}\) be unital and
\(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\). Let \(V(M)\) denote the
affine \(R{\hbox{-}}\)scheme
\(V(M) \coloneqq\operatorname{Spec}\qty{\operatorname{Sym}^\bullet M}\),
which represents
\begin{align*}
S\mapsto \mathop{\mathrm{Hom}}_{{\mathsf{S}{\hbox{-}}\mathsf{Mod}}}(M\otimes_R S, S)
.\end{align*}
We call \(V(M)\) the \textbf{vector group scheme} of \(M\).

\end{definition}

\begin{remark}

Note that \(V({-})\) commutes with arbitrary base change of rings.

\end{remark}

\begin{proposition}[Serre-Swan for Vector Group Schemes]

\(V\) as a functor induces an antiequivalence of categories between
\({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) and vector group schemes:
\begin{align*}
{\mathsf{R}{\hbox{-}}\mathsf{Mod}} &\rightleftharpoons{ \mathsf{Vect} }{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}  \\
M &\mapsto V(M) \\
\Theta(R) &\mapsfrom R
.\end{align*}

\end{proposition}

\begin{remark}

If
\(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}\),
we can consider its dual \(M {}^{ \vee }\). Then
\(\operatorname{Sym}^\bullet M\) is finitely presented, and
\(S\to M\otimes_R S\) is represented by
\(W(M) \coloneqq V(M {}^{ \vee })\). Note that finite locally free is a
necessary condition.

\end{remark}

\hypertarget{vector-bundles-over-affine-schemes}{%
\subsection{Vector Bundles over Affine
Schemes}\label{vector-bundles-over-affine-schemes}}

\begin{definition}[Vector bundle over an affine scheme]

A \textbf{vector bundle of rank \(r\) over the affine scheme
\(\operatorname{Spec}R\)} is an \(X\in {\mathsf{Sch}}_{/ {R}}\) with a
partition \(1 = \sum_i f_i\) along with isomorphisms
\begin{align*}
\varphi_i: V\qty{\qty{R \left[ { \scriptstyle f_i^{-1}} \right]}^r} \to X\underset{\scriptscriptstyle {R} }{\times} R \left[ { \scriptstyle f_i^{-1}} \right]
\end{align*}
where the transitions
\begin{align*}
\varphi_i \varphi_j^{-1}: V\qty{\qty{R \left[ { \scriptstyle (f_i, f_j)^{-1}} \right]}^r} {\circlearrowleft}
\end{align*}
are linear automorphisms.

\end{definition}

\begin{theorem}[Swan-Serre]

\(M\to V(M)\) induces an equivalence between the groupoid of locally
free \(R{\hbox{-}}\)modules of rank \(r\) and the groupoid of vector
bundles over \(\operatorname{Spec}R\) of rank \(r\).
\begin{align*}
{\mathsf{Grpd}}\ni \hspace{4em}
{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r} &\quad{\rightleftharpoons}\quad \underset{\operatorname{Spec}R}{{\mathsf{Bun}}^r} 
\hspace{4em}\in {\mathsf{Grpd}}
.\end{align*}

\end{theorem}

\begin{example}[Tangent Bundles]

Given a smooth\footnote{Since we're in the flat and locally finitely
  presented case, it's sufficient that all fibers are smooth} map of
affine schemes
\begin{align*}
(X\to Y) \coloneqq(\operatorname{Spec}S\to \operatorname{Spec}R)
&&
r\coloneqq\operatorname{reldim}_{X/Y} \geq 1
,\end{align*}
take the tangent bundle, which is dimension \(r\):
\begin{align*}
T_{X/Y} = V(\Omega_{S_{/ {R}} }^1) \in \underset{\operatorname{Spec}R}{\mathsf{Bun}}
.\end{align*}

\end{example}

\begin{example}[Tangent bundle of the real sphere]

Consider the real sphere
\begin{align*}
Z \coloneqq\operatorname{Spec}{\mathbb{R}}[x,y,z] / \left\langle{x^2 + y^2 + z^2}\right\rangle
.\end{align*}
Its tangent bundle \(T_{Z/{\mathbb{R}}}\) is a nontrivial dimension 2
vector bundle, which is classical but can be proven algebraically. As a
consequence, \(Z\) can not be equipped with the structure of a
nontrivial algebraic group over \({\mathbb{R}}\).

\end{example}

\hypertarget{linear-groups}{%
\subsection{Linear Groups}\label{linear-groups}}

\begin{remark}

For
\(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}\),
consider
\begin{align*}
\mathop{\mathrm{End}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M) \xrightarrow{\sim}  M {}^{ \vee }\otimes_R M \in {\mathsf{Alg}}_{/ {R}}  \cap{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}
,\end{align*}
and so
\(V(\mathop{\mathrm{End}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}(M))\)
makes sense. Thus \(V\) is a functor
\begin{align*}
V: \mathsf{Assoc}{\mathsf{Alg}}^{{\operatorname{unital}}} \to{ \mathsf{Vect} }{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}} 
.\end{align*}

Consider \(S\mapsto \mathop{\mathrm{Aut}}_S(M\otimes_R S)\), which is
representable by
\begin{align*}
W(\mathop{\mathrm{End}}_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}} M) \coloneqq\operatorname{GL}(M) = \operatorname{GL}_1(M)
.\end{align*}
Note that there are left and right actions
\begin{align*}
W(M) \curvearrowleft\operatorname{GL}(M)\curvearrowright V(M) 
.\end{align*}

If \(R\) is Noetherian, then
\(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty} \iff \operatorname{GL}(M)\)
is representable. Taking \(M\coloneqq M^r\) recovers the usual
\(\operatorname{GL}_n(R)\) for \(n\coloneqq\operatorname{rank}_R M\).
Note that local freeness is necessary for representability by a group
scheme here.

\end{remark}

\begin{remark}

Given
\(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}\)
there is a partition \(1 = \sum f_i\) and isomorphisms to free
\(R{\hbox{-}}\)modules
\begin{align*}
\varphi_i: R \left[ { \scriptstyle f_i^{-1}} \right]^r \to M\underset{\scriptscriptstyle {R} }{\times} R \left[ { \scriptstyle f_i^{-1}} \right]
&&
\varphi_i^{-1}\varphi_j \leadsto
g_{ij} \in \operatorname{GL}_r\qty{R \left[ { \scriptstyle (f_i, f_j)^{-1}} \right]}
.\end{align*}
These \(g_{ij}\) in fact satisfy a cocycle condition:
\begin{align*}
g_{ij} g_{jk} = g_{ik} \in \operatorname{GL}_r(R \left[ { \scriptstyle f_i f_j f_k^{-1}} \right])
.\end{align*}

\end{remark}

\begin{definition}[Čech Nonabelian Cohomology]

Take an affine cover
\begin{align*}
{\mathcal{U}}\coloneqq\left\{{\operatorname{Spec}R \left[ { \scriptstyle f_i^{-1}} \right]}\right\}_{i\in I}\rightrightarrows\operatorname{Spec}R
\end{align*}
and define \(H^1({\mathcal{U}}_{/ {R}} ; \operatorname{GL}_r)\) to be
1-cocycles up to some notion of cohomological equivalence. This attaches
a vector bundle \(V(M)\) of rank \(r\) a class
\(\gamma(M) \in H^1({\mathcal{U}}_{/ {R}} ; \operatorname{GL}_r)\).
Conversely, by Zariski gluing, for any such cocycle \(g_{ij}\) we can
assign some
\(V_g\in { { {\mathsf{Bun}}_{\operatorname{GL}_r} }}^r_{/ {R}}\) with a
trivializations satisfying \(\varphi_i \varphi_j^{-1}= g_{ij}\).

By taking a limit over all covers, we can define
\begin{align*}
{\check{H}}^1_{\mathrm{zar}}(R; \operatorname{GL}_r) \coloneqq\displaystyle\colim_{{\mathcal{U}}} H^1({\mathcal{U}}_{/ {R}} ; \operatorname{GL}_r)
,\end{align*}
the Čech nonabelian cohomology of \(\operatorname{GL}_r\) with respect
to the Zariski topology on \(\operatorname{Spec}R\).

\end{definition}

\begin{remark}

This classifies \([V] \in {\mathsf{Bun}}_{/ {R}} ^{\cong, r}\) which are
trivialized by \({\mathcal{U}}\). So there are induced maps
\begin{align*}
f: \operatorname{GL}_r \to \operatorname{GL}_s \quad\leadsto\quad f_*: V_r\in {\mathsf{Bun}}^r_{/ {R}}  \to V_s \in {\mathsf{Bun}}^s_{/ {R}} 
\end{align*}
which extend to functors
\begin{align*}
f_*: {\mathsf{Bun}}_{/ {R}} ^r \to {\mathsf{Bun}}_{/ {R}} ^s
.\end{align*}

\end{remark}

\begin{slogan}

Nice constructions for vector bundles arise from morphisms of group
schemes.

\end{slogan}

\hypertarget{classification-of-modules-over-a-dedekind-ring}{%
\subsection{Classification of Modules over a Dedekind
Ring}\label{classification-of-modules-over-a-dedekind-ring}}

\begin{example}[Direct sums]

Write \(n= \sum n_i\) to get a block map
\begin{align*}
f: \prod_{i} \operatorname{GL}_{n_i} &\to \operatorname{GL}_n \\
(A_1, \cdots, A_\ell) &\mapsto A_1 \oplus \cdots \oplus A_\ell
.\end{align*}
In general, the diagonal map obtained by setting \(n_i=1\) for all \(i\)
yields \({\mathbb{G}}_m^{\times r} \to \operatorname{GL}_r\) a
decomposable vector bundle, i.e.~a direct sum of rank 1 bundles.

\end{example}

\begin{example}[Tensor products]

Write \(n= \prod r_i\) to get a similar map, sometimes called the
\emph{Kronecker product}:
\begin{align*}
f: \prod_{i} \operatorname{GL}_{r_i} &\to \operatorname{GL}_n \\
(A_1, \cdots, A_\ell) &\mapsto A_1 \otimes_R \cdots \otimes_R A_\ell
,\end{align*}

\end{example}

\begin{example}[Determinant]

We set
\(\operatorname{det}(V) \coloneqq\operatorname{det}_*(V) \coloneqq\bigwedge\nolimits^r V\),
the determinant bundle.

\end{example}

\begin{remark}

The next result is a classical theorem in commutative algebra, and the
goal is to give a geometric proof.

\end{remark}

\begin{theorem}[Classification of modules over a Dedekind ring]

Let \(R\) be a Dedekind ring \footnote{Noetherian domain where the
  localizations at maximal ideals are DVRs,} , then for any
\begin{align*}
R\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r\geq 1} \implies R \cong R^{r-1} \oplus I && I = \operatorname{det}(R^{r-1} \oplus I) \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\times}, 
\end{align*}
where \(I\) is invertible and unique up to isomorphism. Thus vector
bundles over \(R\) are decomposable and classified by their determinant.

\end{theorem}

\begin{corollary}[Characterization of trivial modules]

A locally free \(R{\hbox{-}}\)module \(M\) of rank \(r\geq 1\) is
trivial \(\iff \operatorname{det}M\) is trivial.

\end{corollary}

\begin{proof}[of the classification theorem]

We're given \(V(M)\) a vector bundle, which trivializes over an affine
subset \(\operatorname{Spec}R \left[ { \scriptstyle f_i^{-1}} \right]\).
Set
\begin{align*}
\Sigma \coloneqq\operatorname{Spec}R / \operatorname{Spec}R \left[ { \scriptstyle f_i^{-1}} \right] = \left\{{{\mathfrak{p}}_i}\right\}_{i=1}^c && {\mathfrak{p}}_i \in \operatorname{mSpec}R
.\end{align*}
Let \(\widehat{R} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]\)
be the completion of the residue DVR and let
\(\widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right] \coloneqq K\otimes_R \widehat{R} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]\)
its fraction field. By Nakayama,
\(M\otimes_R \widehat{R} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]\)
is free, so pick a trivialization. We can use this to produce a double
coset:
\begin{align*}
c_{\Sigma}\left(R; \operatorname{GL}_{r}\right)
\coloneqq
\operatorname{GL}_{r}\left(R_{f}\right) 
\backslash 
\quad \prod_{j=1}^c \operatorname{GL}_{r}
\left(
\widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]
\right) 
\quad / 
\operatorname{GL}_{r}
\left(
\widehat{R}  \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]
\right)
.\end{align*}

\begin{claim}

The following map is injective:
\begin{align*}
\ker\qty{H^1(R; \operatorname{GL}_r) \to H^1\qty{R \left[ { \scriptstyle f^{-1}} \right]; \operatorname{GL}_r}} \to c_\Sigma(R; \operatorname{GL}_r)
,\end{align*}
although we'll only need that its kernel is trivial.

\end{claim}

We can assume \(\operatorname{det}V(M)\) is trivial to get
\begin{align*}
g_i \in \ker \operatorname{det}_* \coloneqq\ker\qty{c_\Sigma(R; \operatorname{GL}_r) \to c_\Sigma(R; {\mathbb{G}}_m)}
.\end{align*}
Changing trivializations, we can assume
\(g_i \in {\operatorname{SL}}_n(\widehat{K}_{{\mathfrak{p}}_i})\), which
is generated by elementary matrices. Using that
\(R_f \subseteq \prod_i \widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]\)
is dense, we get
\begin{align*}
{\operatorname{SL}}_n\qty{\widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]} \subseteq \prod_i {\operatorname{SL}}_n \qty{ \widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]} \text{ is dense}
.\end{align*}
But since each
\({\operatorname{SL}}_n\qty{\widehat{R} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]}\)
is clopen in
\({\operatorname{SL}}_n\qty{\widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]}\),
we obtain \(c_\Sigma(R; {\operatorname{SL}}_r) = 1\) and injectivity
allows us to conclude that \(V(M)\) is a trivial vector bundle.

\end{proof}

\begin{remark}

This is a ``strong approximation'' argument.

\end{remark}

\hypertarget{replacing-the-zariski-topology}{%
\subsection{Replacing the Zariski
Topology}\label{replacing-the-zariski-topology}}

\begin{definition}[Quadratic Forms]

Given an \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), a map \(q:M\to R\)
is a \textbf{quadratic form} iff

\begin{itemize}
\tightlist
\item
  \(q(\lambda x) = \lambda^2 q(x)\) for all \(\lambda\in R, x\in M\),
\item
  The associated form is symmetric and bilinear:
  \begin{align*}
  b_q: M\otimes_R M &\to R \\
  (x, y) &\mapsto q(x+y) - q(x) -q(y)
  .\end{align*}
\end{itemize}

The form \(q\) is \textbf{regular} iff \(b_q\) induces an isomorphism
\(M \xrightarrow{\sim} M {}^{ \vee }\).

\end{definition}

\begin{example}[The hyperbolic form]

For
\(V\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, \operatorname{rank}< \infty}\),
the \textbf{hyperbolic form} is defined by
\begin{align*}
q_{\operatorname{hyp}}: V \oplus V {}^{ \vee }&\to R \\
v \otimes\psi &\mapsto \psi(v)
.\end{align*}

\end{example}

\begin{remark}

The definition of nonabelian cohomology will extend to arbitrary group
schemes, but the Zariski topology is not fine enough. One reason to try
extending this theory will be quadratic bundles. For a regular quadratic
form \((M, q)\) with
\(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, \operatorname{rank}< \infty}\),
we can draw an analogy with usual quadratic forms. So for any
\(R{\hbox{-}}\)ring \(S\) we could define a subgroup scheme
\begin{align*}
{\operatorname{O}}(q, M) \coloneqq\left\{{g \in \operatorname{GL}(M)(S) {~\mathrel{\Big|}~}q_S \circ g = q_S }\right\} \leq \operatorname{GL}(M)
\end{align*}
and similarly define
\(H^1({\mathcal{U}}_{/ {R}} ; {\operatorname{O}}(q, M))\) for any open
cover \({\mathcal{U}}\rightrightarrows R\).

\end{remark}

\begin{lemma}[?]

\(H^1_{\mathrm{zar}}({\mathcal{U}}_{/ {R}} ; {\operatorname{O}}(q, M))\)
classifies isomorphism classes of regular quadratic forms \((q', M')\)
which are locally isomorphic over \({\mathcal{U}}\) to \((q, M)\).

\end{lemma}

\begin{remark}

Upshot: not all regular quadratic forms over \(R\) of a fixed dimension
\(r\) need be locally isomorphic, noting that this already fails for
\(R\coloneqq{\mathbb{R}}\).

\end{remark}

\begin{remark}

Given a morphism of group schemes \(f:G\to H\), we would like control
over \(H^1_{\mathrm{zar}}(R; G) \to H^1_{\mathrm{zar}}(R; H)\). Consider
the Kummer map
\begin{align*}
f_d: {\mathbb{G}}_m &\to {\mathbb{G}}_m \\
t &\mapsto t^d
.\end{align*}
This induces \(\times d\) on \({\operatorname{Pic}}(R)\), and on
\({\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\times}\) corresponds to
\(M\to M^{\otimes d}\). We'd like to understand its kernel and image,
which will generally involve higher \(H_\text{ét}\). Given
\begin{align*}
[M] \in \ker ({\operatorname{Pic}}(R) \xrightarrow{\times d} {\operatorname{Pic}}(R))
,\end{align*}
there is a trivialization \(\theta: R\to M^{\otimes d}\). We'll define a
group
\begin{align*}
A_d(R) \coloneqq\left\{{ (M, \theta) {~\mathrel{\Big|}~}M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\times}, \theta \text{ a trivialization}}\right\}/\cong
,\end{align*}
which will correspond to something in \(H_\text{ét}\). There is an exact
sequence

\begin{center}
\begin{tikzcd}
    {R^{\times}/(R^{\times})^d} && {A_d(R)} && {{\operatorname{Pic}}(R)} && {{\operatorname{Pic}}(R)} \\
    a && {[R, {\theta_a: R\to R^{\otimes d}, x\mapsto ax} ]} \\
    && {[M, \theta]} && {[M]}
    \arrow["\varphi", from=1-1, to=1-3]
    \arrow["\mathop{\mathrm{Forget}}", from=1-3, to=1-5]
    \arrow["{\times d}", from=1-5, to=1-7]
    \arrow[maps to, from=2-1, to=2-3]
    \arrow[maps to, from=3-3, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMCwwLCJSXFx1bml0cy8oUlxcdW5pdHMpXmQiXSxbMiwwLCJBX2QoUikiXSxbNCwwLCJcXFBpYyhSKSJdLFs2LDAsIlxcUGljKFIpIl0sWzAsMSwiYSJdLFsyLDEsIltSLCB7XFx0aGV0YV9hOiBSXFx0byBSXntcXHRlbnNvciBkfSwgeFxcbWFwc3RvIGF4fSBdIl0sWzIsMiwiW00sIFxcdGhldGFdIl0sWzQsMiwiW01dIl0sWzAsMSwiXFxwaGkiXSxbMSwyLCJcXEZvcmdldCJdLFsyLDMsIlxcdGltZXMgZCJdLFs0LDUsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbNiw3LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV1d}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{remark}

Grothendieck-Serre's idea is to extend the notion of covers, first with
étale covers, and later with flat covers which are simpler as a first
approach.

\end{remark}

\begin{definition}[Flat Covers]

A \textbf{flat (fppf) cover} of \(R\) is a finite collection
\(\left\{{S_i}\right\}\) of \(R{\hbox{-}}\)rings where \(S_i\) is a flat
finitely presented \(R{\hbox{-}}\)algebra and
\begin{align*}
\operatorname{Spec}R = \displaystyle\bigcup_{i\in I} \operatorname{im}(\operatorname{Spec}S_i \to \operatorname{Spec}R)
.\end{align*}

\end{definition}

\begin{remark}

Setting \(S\coloneqq{\textstyle\coprod}S_i\), this says \(S\) is a
faithfully flat finitely presented \(R{\hbox{-}}\)algebra. Note that the
Zariski cover is a flat cover.

\end{remark}

\begin{remark}

For \(G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {R}}\), a 1-cocycle for \(G\)
and \(S_{/ {R}}\) is an element \(g\in G(S\otimes_R S)\) where
\begin{align*}
q_{12}^*(g) g_{2,3}^*(g) = q_{1, 3}^* \in G(S^{\otimes_R 3})
.\end{align*}
We can use the finite presentation hypothesis to pass to a limit over
all flat covers of \(\operatorname{Spec}R\) and define
\begin{align*}
H^1_\mathrm{\operatorname{fppf}}(R; G)\coloneqq\colim H^1(S_{/ {R}} ; G)
.\end{align*}
.

\end{remark}

\begin{definition}[Torsors]

A \textbf{right \(G{\hbox{-}}\)torsor}
\(X\in {\mathsf{Sch}}_{/ {R}} \curvearrowleft G\), so \(X\) is a scheme
over \(R\) with a right \(G{\hbox{-}}\)action, where

\begin{itemize}
\item
  The following map is an isomorphism:
  \begin{align*}
  X { \underset{\scriptscriptstyle {R} }{\times} } G &\xrightarrow{\sim} X{ \underset{\scriptscriptstyle {R} }{\times} } {\color{green} X} \\
  (x, g) &\mapsto (x, x\cdot g)
  .\end{align*}
  This says that \(G(T) \curvearrowright X(T)\) for all
  \(T\in \mathsf{Ring}_{/ {R}}\).
\item
  There exists a flat cover
  \(\left\{{R_i{}_{/ {R}} }\right\}_i \rightrightarrows X\) with
  \(X(R_i)\neq 0\).

  This says \(X\) is locally trivial in the flat topology.
\end{itemize}

Morphisms of torsors are \(G{\hbox{-}}\)equivariant maps of schemes, and
condition 1 forces these to all be isomorphisms, so
\({\mathsf{Torsor}{\hbox{-}}\mathsf{G} }_{/ {R}} \in {\mathsf{Grpd}}\).

\end{definition}

\begin{definition}[Split torsor]

Setting \(X\coloneqq G\) with \(G\curvearrowright G\) by
right-translation yields the \textbf{split torsor}.

\end{definition}

\begin{definition}[Trivial Torsor]

If \(X(R) \neq \emptyset\), so \(X\) has an \(R{\hbox{-}}\)point, the
point \(x\in X(R)\) defines an isomorphism
\begin{align*}
G &\xrightarrow{\sim} X \\
g &\mapsto x\cdot g
.\end{align*}

In this case, we say that \(X\) is a \textbf{trivial torsor}.

\end{definition}

\begin{example}[?]

\(G\in {\mathsf{G}{\hbox{-}}\mathsf{Torsor}}_{/ {R}}\) is a trivial
\(G{\hbox{-}}\)torsor.

\end{example}

\begin{definition}[First fppf cohomology]

The functor
\begin{align*}
R\mapsto \mathop{\mathrm{Aut}}_{{\mathsf{Torsor}{\hbox{-}}\mathsf{G} }_{/ {R}} }(G)
.\end{align*}
of automorphisms of the trivial \(G{\hbox{-}}\)torsor \(G\) is
representable by \(G\), acting by left translation. We formally define
the \textbf{first fppf cohomology} to be isomorphism classes of
\(G{\hbox{-}}\)torsors:
\begin{align*}
H^1_\mathrm{\operatorname{fppf}}(R; G) \coloneqq{\mathsf{Torsor}{\hbox{-}}\mathsf{G} }^{\cong}_{/ {R}}  \text{ for the flat topology}
,\end{align*}
and for \(S\to R \in {\mathsf{Cov}}^\mathrm{flat}(R)\), we define the
subset of \(G{\hbox{-}}\)torsors trivialized over \(S\) as
\begin{align*}
H^1_\mathrm{\operatorname{fppf}}(S_{/ {R}} ; G) \subseteq H^1_\mathrm{\operatorname{fppf}}(R; G)
.\end{align*}

\end{definition}

\begin{definition}[Class map]

There is a \textbf{class map}
\begin{align*}
\gamma: H^1_\mathrm{\operatorname{fppf}}(S_{/ {R}} ; G) \to {\check{H}}^1_\mathrm{\operatorname{fppf}}(S_{/ {R}} ; G)
.\end{align*}

\end{definition}

\todo[inline]{How do you construct the class map..?}

\begin{remark}

For \(X\in {\mathsf{G}{\hbox{-}}\mathsf{Torsor}}_{/ {R}}\) with a
trivialization
\begin{align*}
\varphi: G{ \underset{\scriptscriptstyle {R} }{\times} } S \xrightarrow{\sim} X { \underset{\scriptscriptstyle {R} }{\times} } S
,\end{align*}
there are two trivializations over \(S^{\otimes_R 2}\):

\begin{center}
\begin{tikzcd}
    {G{ \underset{\scriptscriptstyle {R} }{\times} } S^{\otimes_R 2}} \\
    \\
    {G{ \underset{\scriptscriptstyle {R} }{\times} } S^{\otimes_R 2}} && {X{ \underset{\scriptscriptstyle {R} }{\times} } S^{\otimes_R 2}}
    \arrow["{\varphi \otimes 1}"', from=3-1, to=3-3]
    \arrow["{p_1^*(\varphi) }"', shift right=1, from=1-1, to=3-3]
    \arrow["{p_2^*(\varphi) = \varphi \otimes 1}", shift left=1, from=1-1, to=3-3]
    \arrow["{1\otimes\tau_{12}}"', from=1-1, to=3-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJHXFxmaWJlcnByb2R7Un0gU157XFx0ZW5zb3JfUiAyfSJdLFswLDIsIkdcXGZpYmVycHJvZHtSfSBTXntcXHRlbnNvcl9SIDJ9Il0sWzIsMiwiWFxcZmliZXJwcm9ke1J9IFMiXSxbMSwyLCJcXHZhcnBoaSBcXHRlbnNvciAxIiwyXSxbMCwyLCJwXzFeKihcXHZhcnBoaSkgIiwyLHsib2Zmc2V0IjoxfV0sWzAsMiwicF8yXiooXFx2YXJwaGkpID0gXFx2YXJwaGkgXFx0ZW5zb3IgMSIsMCx7Im9mZnNldCI6LTF9XSxbMCwxLCIxXFx0ZW5zb3IgXFx0YXVfezEyfSIsMl1d}{Link
to Diagram}
\end{quote}

Thus
\(p_1(\varphi)^{-1}\circ p_2(\varphi) \in \mathop{\mathrm{Aut}}_{{\mathsf{G}{\hbox{-}}\mathsf{Torsor}}}(G)_{/ {S^{\otimes_R 2} }}\)
is an automorphism of the trivial torsor, thus acts by left translation
by some \(g \in G(S^{\otimes_R 2})\). An argument shows that \(g\) is a
1-cocycle and that changing \(\varphi\) only changes \(g\) by a
coboundary, so the class map is well-defined.

\end{remark}

\begin{definition}[Amitsur Resolution]

If \(X\in {\mathsf{G}{\hbox{-}}\mathsf{Torsor}}\) has a trivialization
\(\phi\), then over \(S^{\otimes_R 2}\) we have two trivializations:

Let \(T\) be a faithfully flat descent of \(R\), then the
\textbf{Amitsur complex} is defined as
\begin{align*}
0\to M\to M\otimes_R T \to M \otimes_R T^{\otimes_R 2} \to \cdots && \leadsto M \to M\otimes_R {\mathbb{T}}^\bullet(T)
,\end{align*}
where \({\mathbb{T}}\) denotes the tensor algebra. This has a
differential given by
\begin{align*}
{{\partial}}(m \otimes\mathbf{t}) = \sum_{i} (-1)^i m \otimes\psi_i(\mathbf{t})
,\end{align*}
where \(\psi_i\) is the \(i\)th face map inserting a 1 between the \(i\)
and \(i+1\)st tensor factors.

\end{definition}

\begin{remark}

An important theorem is that the Amitsur complex is exact for each
\(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), which for any
\(X\in {\mathsf{Aff}}{\mathsf{Sch}}_{/ {R}}\) allows an identification
\begin{align*}
X(R) = \left\{{ x\in X(T) {~\mathrel{\Big|}~}p_1^*(x) = p_2^*(x) \in X\qty{ T^{\otimes_R 2} }  }\right\} 
.\end{align*}

\end{remark}

\begin{definition}[Descent data]

For \(N\in {\mathsf{T}{\hbox{-}}\mathsf{Mod}}\), consider
\begin{align*}
p_1^*(N) \coloneqq T\otimes_R M
,\quad
p_2^*(N) \coloneqq M\otimes_R T
 && 
 \in \mathsf{T^{\otimes_R 2}}{\hbox{-}}\mathsf{Mod}
.\end{align*}

A \textbf{descent datum} on \(N\) is an isomorphism
\begin{align*}
\varphi: p_1^*(N) \xrightarrow{\sim} p_2^*(N) && \in \mathsf{T^{\otimes_R 2}}{\hbox{-}}\mathsf{Mod}
.\end{align*}
of \(T^{\otimes_R 2}{\hbox{-}}\)modules making the following diagram
commute:

\begin{center}
\begin{tikzcd}
    {T^{\otimes_R 2} \otimes_R N} &&&& {N \otimes_R T^{\otimes_R 2}} \\
    \\
    \\
    && {T \otimes_R N \otimes_R T}
    \arrow["{\varphi_2 = (1\otimes\varphi)\circ \tau_{12}}", from=1-1, to=4-3]
    \arrow["{\varphi_3 = (\varphi\otimes 1) \circ \tau_{23}}"', from=1-1, to=1-5]
    \arrow["{\varphi_1 = 1\otimes\varphi}"', from=1-5, to=4-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJUXntcXHRlbnNvcl9SIDJ9IFxcdGVuc29yX1IgTiJdLFs0LDAsIk4gXFx0ZW5zb3JfUiBUXntcXHRlbnNvcl9SIDJ9Il0sWzIsMywiVCBcXHRlbnNvcl9SIE4gXFx0ZW5zb3JfUiBUIl0sWzAsMiwiXFx2YXJwaGlfMiA9ICgxXFx0ZW5zb3IgXFx2YXJwaGkpXFxjaXJjIFxcdGF1X3sxMn0iXSxbMCwxLCJcXHZhcnBoaV8zID0gKFxcdmFycGhpXFx0ZW5zb3IgMSkgXFxjaXJjIFxcdGF1X3syM30iLDJdLFsxLDIsIlxcdmFycGhpXzEgPSAxXFx0ZW5zb3IgXFx2YXJwaGkiLDJdXQ==}{Link
to Diagram}
\end{quote}

Here \(\tau_{ij}\) is the map that swaps the \(i\) and \(j\)th tensor
factors, so
e.g.~\(\varphi_3(t_1 \otimes t_2 \otimes n) \coloneqq\varphi(t_1\otimes n)\otimes t_2\).

There is a category of \(T{\hbox{-}}\)modules with descent data, where
objects are pairs \((T, \varphi)\) and morphisms are clear, and I'll
write this as \({\mathsf{T}{\hbox{-}}\mathsf{Mod}}{\mathsf{Desc}}\). For
\(M\in{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), there is a \textbf{canonical
descent datum}
\begin{align*}
{\mathrm{can}}_M: p_1^*(M\otimes_R T) \xrightarrow{\sim} p_2^*(M\otimes_R T)
.\end{align*}

\end{definition}

\begin{theorem}[(Grothendieck) Faithfully flat descent]

There is a functor inducing an equivalence of categories:
\begin{align*}
F: {\mathsf{R}{\hbox{-}}\mathsf{Mod}}&\rightleftharpoons{\mathsf{T}{\hbox{-}}\mathsf{Mod}}{\mathsf{Desc}}\\
M &\mapsto (M\otimes_R T, {\mathrm{can}}_M) \\
\left\{{ n\in N {~\mathrel{\Big|}~}n\otimes 1 = \varphi(1\otimes n) }\right\} 
&\mapsfrom
(N, \varphi) 
.\end{align*}
This induces an equivalence of categories
\begin{align*}
{\mathsf{cAlg}}_{/ {R}} ^{{\operatorname{unital}}} &\rightleftharpoons{\mathsf{cAlg}}^{{\operatorname{unital}}}_{/ {T}} {\mathsf{Desc}}
.\end{align*}

\end{theorem}

\begin{remark}

This theorem is a good reason to focus on the affine setting. Faithfully
flat descent implies the following:

\end{remark}

\begin{theorem}[?]

Let
\(M \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r}\)
then the functor
\begin{align*}
\mathsf{S}{\hbox{-}}\mathsf{Mod} &\to {\mathsf{Set}}\\
S &\mapsto \mathop{\mathrm{Isom}}_{{\mathsf{S}{\hbox{-}}\mathsf{Mod}}}(S^r, M\otimes_R S)
\end{align*}
is representable by an object \(X^M\). This induces an equivalence of
categories
\begin{align*}
{\mathsf{Grpd}}\ni \hspace{4em} X^{({-})}: {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r} &\rightleftharpoons{\mathsf{\operatorname{GL}_r}{\hbox{-}}\mathsf{Torsor}}_{/ {R}}  \hspace{4em} \in {\mathsf{Grpd}}
.\end{align*}

\end{theorem}

\begin{corollary}[Hilbert=Grothendieck 90]

\begin{align*}
H^1_{\mathrm{zar}}(R; \operatorname{GL}_r) \cong H^1_\mathrm{\operatorname{fppf}}(R; \operatorname{GL}_r)
.\end{align*}

\end{corollary}

\begin{remark}

In particular, if \(R\) is local or semilocal,
\(H^1_\mathrm{\operatorname{fppf}}(R; \operatorname{GL}_r) = 1\). This
also holds for \(R\) replaced by any
\(B \in {\mathsf{Alg}}_{/ {R}} { {}^{ \operatorname{sep} } }\), e.g.~a
finite étale or Azumaya algebra.

\end{remark}

\begin{lemma}[The class map is injective]

\begin{align*}
H^1_\mathrm{\operatorname{fppf}}(S_{/ {R}} ; G)\hookrightarrow{\check{H}}^1_\mathrm{\operatorname{fppf}}(S_{/ {R}} ; G)
.\end{align*}

\end{lemma}

\begin{remark}

By passing to the limit over flat covers, we get an isomorphism on
\(H^1_\mathrm{\operatorname{fppf}}(R; G) \to {\check{H}}^1_\mathrm{\operatorname{fppf}}(R; G)\),
and we can descend torsors under an affine scheme. The proof involves
the following construction:

\end{remark}

\begin{definition}[Twisting]

If \(Y \in G\curvearrowright{\mathsf{Aff}}{\mathsf{Sch}}_{/ {R}}\) has a
left \(G{\hbox{-}}\)action, then the action map
\begin{align*}
G\underset{\scriptscriptstyle {R} }{\times} S^{\otimes_R 2}
\xrightarrow{\sim}
Y\underset{\scriptscriptstyle {R} }{\times} S^{\otimes_R 2}
\end{align*}
is an isomorphism that defines a descent datum. If \(G\) acts on itself
by inner automorphism, \(G_g\) is called that twisted
\(R{\hbox{-}}\)group scheme, which acts on \(Y_g\). So for any
\(E\in {\mathsf{G}{\hbox{-}}\mathsf{Torsor}}\), we can define the
\textbf{twists} \({ {}^{Y} E }\) and \({ {}^{G} E }\). In general, we
can twist \(G{\hbox{-}}\)schemes equipped with an amply invertible
\(G{\hbox{-}}\)linearized line bundle.

\end{definition}

\begin{theorem}[?]

If \(G\) is affine, the class map \(\gamma\) is an isomorphism.

\end{theorem}

\hypertarget{phillippe-gille-talk-2-tuesday-july-13}{%
\section{Phillippe Gille, Talk 2 (Tuesday, July
13)}\label{phillippe-gille-talk-2-tuesday-july-13}}

\hypertarget{intro-2}{%
\subsection{Intro}\label{intro-2}}

\begin{remark}

Let \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\). Reminders of notation

\begin{itemize}
\tightlist
\item
  \(V(M) \coloneqq\operatorname{Spec}\operatorname{Sym}^\bullet M\)
  which represents
  \begin{align*}
  S \mapsto \hom_{{\mathsf{(S)}{\hbox{-}}\mathsf{Mod}}}(M\otimes_R S, S)
  ,\end{align*}
  is the \textbf{vector group scheme} of \(M\).
\item
  \(W(M)\coloneqq V(M {}^{ \vee })\) which represents \(M\otimes_R {-}\)
  (and doesn't seem to have a name).
\item
  For \(Y\in G{\hbox{-}}{\mathsf{Aff}}{\mathsf{Sch}}_{/R}\) (affine
  schemes with a left \(G{\hbox{-}}\)action),
  \(Y_g\in {\mathsf{Aff}}{\mathsf{Sch}}_{/R}\) is the \textbf{twist} of
  \(Y\) by the 1-cocycle \(g\) defined by the action map:
  \begin{align*}
  g: 
  Y \underset{\scriptscriptstyle {R} }{\times} S^{\otimes_R 2} 
  \xrightarrow{\sim} 
  Y \underset{\scriptscriptstyle {R} }{\times} S^{\otimes_R 2} 
  .\end{align*}

  \begin{itemize}
  \tightlist
  \item
    For any \(G{\hbox{-}}\)torsor \(E\) and any
    \(Y\in G{\hbox{-}}{\mathsf{Sch}}\) with an ample invertible
    \(G{\hbox{-}}\)linearized bundle\footnote{This holds for example if
      \(Y \in {\mathsf{Aff}}{\mathsf{Sch}}\).} , one can similarly
    define \textbf{twists} \({ {}^{E} Y }\).
  \end{itemize}
\item
  For \(T\) a faithfully flat extension of \(R\), the \textbf{Amitsur
  resolution} is given by
  \begin{align*}
  M\to M \otimes_R {\mathbb{T}}(T)^\bullet
  && \text{where }
  {\mathbb{T}}(V)^\bullet \coloneqq V \oplus V^{\otimes 2} \oplus \cdots
  .\end{align*}
  I.e., this resolves \(M\) by the tensor algebra (or free algebra) on
  \(T\).
\end{itemize}

We'll now discuss some important special cases of
\(G{\hbox{-}}\)torsors. The following claim is in analogy to
\({\mathsf{Coh}}(X)\) for \(X\in {\mathsf{Aff}}{\mathsf{Sch}}\):

\end{remark}

\begin{fact}[Vector group schemes have trivial cohomology]

If
\(M\in {\mathsf{Fin}}{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r<\infty}\),
then \({\check{H}}^1(R; W(M)) = 0\) and every \(W(M){\hbox{-}}\)torsor
is trivial.

The following are some important special cases:

\end{fact}

\begin{definition}[Finite Constant Group Scheme]

For any \(\Gamma\in {\mathsf{Fin}}{\mathsf{Grp}}\), \(\Gamma_R\) is the
\textbf{finite constant group scheme} attached to any \(\Gamma\), and is
defined by
\begin{align*}
\Gamma_R(S) \coloneqq\left\{{ f:\operatorname{Spec}S\to \Gamma {~\mathrel{\Big|}~}f\text{ is locally constant} }\right\} \in {\mathsf{Grp}}
.\end{align*}

\end{definition}

\hypertarget{galois-covers}{%
\subsection{Galois Covers}\label{galois-covers}}

\begin{definition}[Galois Cover]

A \(\Gamma_R{\hbox{-}}\)torsor of the form
\(\operatorname{Spec}S\to \operatorname{Spec}R\) is equivalently a
Galois \(\Gamma{\hbox{-}}\)algebra \(S\) and is referred to as a
\textbf{Galois cover}.

\end{definition}

\begin{example}[?]

A finite Galois extension \(L_{/ {k}}\) with Galois group
\(\Gamma \coloneqq{ \mathsf{Gal}} (L_{/ {k}} )\) yields a Galois cover
\(\operatorname{Spec}L\to \operatorname{Spec}k\).

\end{example}

\begin{example}[?]

Another nice case is when \(\Gamma\) is the automorphism group
associated to some algebraic structure, i.e.~when one has
\textbf{forms}. For example, take
\(\Gamma \coloneqq{\operatorname{O}}_{2n} = \mathop{\mathrm{Aut}}(q_{\operatorname{hyp}})\)
for the hyperbolic form \(q_{\operatorname{hyp}}\) on \(R^n\). Descent
gives an equivalence of categories
\begin{align*}
{\mathsf{Grpd}}\ni \left\{{\substack{ \text{Regular quadratic forms } q \\ \text{with } \operatorname{rank}q = 2n}}\right\} 
\rightleftharpoons
H^1_{\mathrm{\operatorname{fppf}}}(R; {\operatorname{O}}_{2n})
,\end{align*}
which uses that all forms appearing on the left-hand side are locally
isomorphic to \(q_{\operatorname{hyp}}\) in the flat topology.

\end{example}

\begin{example}[?]

Take \(\Gamma \coloneqq S_n\) the symmetric group, so
\begin{align*}
S_n(X) = \mathop{\mathrm{Aut}}_{\mathsf{Grp}}(X^{\times n}) && \forall S \in {\mathsf{Alg}}_{/R}
.\end{align*}

The same yoga shows there is a categorical equivalence
\begin{align*}
S_n{\hbox{-}}\text{torsors} &\rightleftharpoons{\mathsf{Fin}}{\mathsf{Alg}}^{\text{ét}}_{/R}
,\end{align*}
where we use that every
\(X\in {\mathsf{Fin}}{\mathsf{Alg}}^{\text{ét}}\) of degree \(n\) is
locally isomorphism to \(R^n\). The inverse is given by descent.

\end{example}

\hypertarget{flat-quotients}{%
\subsection{Flat Quotients}\label{flat-quotients}}

\begin{definition}[Flat Quotient]

For \(X, H\in {\mathsf{Grp}}{\mathsf{Sch}}_{/R}\), a map \(H\to X\) is a
\textbf{flat quotient} of \(H\) by \(G\) iff

\begin{itemize}
\tightlist
\item
  For each \(S\in {\mathsf{Alg}_{/R}}\) the map \(H(S)\to X(S)\) induces
  an injection \(H(S)/G(S)\hookrightarrow X(S)\), and
\item
  For each \(x\in X(S)\) there exists a flat cover \(S'\to S\) with
  \(x_{S'} \in \operatorname{im}(H(S') \to X(S'))\) (so \(f\) is
  \emph{couvrant} en français).
\end{itemize}

\end{definition}

\begin{example}[of flat quotients]

\envlist

\begin{itemize}
\tightlist
\item
  \({\mathbb{G}}_m = \operatorname{GL}_r /_{{\operatorname{Flat}}}\, {\operatorname{SL}}_r\).
\item
  \({\mathbb{G}}_m = {\mathbb{G}}_m /_{{\operatorname{Flat}}}\, \mu_d\).
\end{itemize}

\end{example}

\begin{lemma}[?]

Let \(X\) be the flat quotient of \(H\) by \(G\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(H\to X\) is a \(G{\hbox{-}}\)torsor
\item
  There is an exact sequence of pointed sets:
  \begin{align*}
  1 \to G(R) \to H(R) \to X(R) \xrightarrow{\phi(x) = [f^{-1}(x)]} H_\mathrm{\operatorname{fppf}}^1(R; G) \to H_\mathrm{\operatorname{fppf}}^1(R; H) 
  ,\end{align*}
  where \(\phi\) is denoted the \emph{characteristic map}, which arises
  naturally from base change.
\end{enumerate}

\end{lemma}

\begin{remark}

When \(G{~\trianglelefteq~}H\), then
\(X\in {\mathsf{Grp}}{\mathsf{Sch}}_{/R}\), and
\begin{align*}
\left\{{\substack{G{\hbox{-}}\text{torsors over }\operatorname{Spec}R }}\right\}
\rightleftharpoons
\left\{{\substack{(F, \phi) {~\mathrel{\Big|}~}F\in H{\hbox{-}}\text{torsors}, \\ \phi \text{ a local trivialization of } { {}^{F} X } }}\right\}
.\end{align*}

\end{remark}

\begin{example}[?]

\({\operatorname{SL}}_n{\hbox{-}}\)torsors are equivalent to
\((M, \theta)\) with
\(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\mathsf{loc}}\mathsf{Free}, r}\)
and \(\theta: R \xrightarrow{\sim} \bigwedge\nolimits^r(M)\) is a
trivialization of \(\operatorname{det}(M)\).

\end{example}

\begin{example}[?]

Using the Kummer exact sequence
\begin{align*}
1 \to \mu_d \to {\mathbb{G}}_m \xrightarrow{\times d} {\mathbb{G}}_m \to 1
,\end{align*}
\(\mu_d{\hbox{-}}\)torsors are equivalent to pairs \((M, \theta)\) with
\(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\times}\) and
\(\theta: R \xrightarrow{\sim} M^{\otimes_R m}\) is a trivialization.

\end{example}

\begin{definition}[étale morphism]

An \textbf{étale morphism} of rings \(R\to S\) is a smooth morphism of
\(\operatorname{reldim}= 0\). Alternatively,
\(S\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{{\operatorname{Flat}}}\) such
that for every \(R{\hbox{-}}\)field \(F\),
\(S\otimes_R F \in {\mathsf{Alg}}_{/F}^{\text{ét}}\), where étale
algebras are finite and geometrically reduced.

\end{definition}

\begin{example}[?]

\begin{itemize}
\tightlist
\item
  Localization \(R\to R \left[ { \scriptstyle f^{-1}} \right]\) is
  étale.
\item
  If \(d^{-1}\in R\) then the Kummer morphism \(t\mapsto t^d\) is étale.
\item
  If \(d^{-1}\in R\) and \(r\in R^{\times}\), then
  \(R[x] /\left\langle{x^d-r}\right\rangle\in {\mathsf{Fin}}{\mathsf{Alg}}_{/R}^\text{ét}\).
\end{itemize}

\end{example}

\begin{proposition}[?]

For \(G\) affine smooth, there is an equivalence of torsors
\begin{align*}
H^1_\text{ét}(R; G) \cong H^1_\mathrm{\operatorname{fppf}}(R;G)
.\end{align*}

\end{proposition}

\begin{proof}[?]

See notes. This uses that in the flat topology, smoothness is local.

\end{proof}

\hypertarget{galois-cohomology-1}{%
\subsection{Galois Cohomology}\label{galois-cohomology-1}}

\begin{remark}

Galois descent is a special case of faithfully flat descent, and takes
the form
\begin{align*}
{\mathsf{R}{\hbox{-}}\mathsf{Mod}}\rightleftharpoons\left\{{\substack{(N, \rho) {~\mathrel{\Big|}~}N\in {\mathsf{S}{\hbox{-}}\mathsf{Mod}} \\ \phi \text{ a semilinear action } \Gamma\curvearrowright N }}\right\}
,\end{align*}
where \textbf{semilinear} means that
\begin{align*}
\varphi(\sigma) (\lambda \cdot n) = \sigma(\lambda) \cdot \rho( \sigma)(n)
.\end{align*}

\end{remark}

\begin{definition}[Isotrivial torsors (very important!)]

A torsor \(X\) over \({\mathsf{Grp}}{\mathsf{Sch}}_{/R}\) is
\textbf{isotrivial} if it is split (trivialized) by a finite Galois
étale cover \(R'\to R\).

\end{definition}

\begin{remark}

These are the torsors that can be made explicit in Galois cohomology.

\end{remark}

\begin{example}[?]

Let \(\operatorname{ch}k=0\), then the ring of Laurent polynomials
\(k{\left(\left( x \right)\right) }\), this is isotrivial and a
reductive group scheme. A special case is that of \textbf{loop torsors},
which are closely related to representation theory in
\({\mathsf{Alg}}{\mathsf{Grp}}\).

\end{example}

\begin{remark}

The main topic is affine curves, and these are special cases of Dedekind
rings. Let \(R\) be Dedekind with \(K\coloneqq\operatorname{ff}(R)\) and
\(G\in {\mathsf{Aff}}{\mathsf{Grp}}{\mathsf{Sch}}\), then as in the
proof for \(\operatorname{GL}_n\) yesterday we have an injective class
map
\begin{align*}
\ker(H^1_\mathrm{\operatorname{fppf}}\to H^1_\mathrm{\operatorname{fppf}}\times \cdots ) \to c_{\Sigma}(\cdots)
.\end{align*}
In particular, if \(c_\Sigma(R; G) =1\), and in particular
\(G(R \left[ { \scriptstyle f^{-1}} \right])\) is dense in
\(\prod \cdots\), the kernel appearing here is trivial.

\end{remark}

\begin{corollary}[?]

If \(G\) is a semisimple simply connected \footnote{Here ``simply
  connected'' is in the sense of semisimple algebraic groups or group
  schemes, and over \({\mathbb{C}}\) coincides with the topological
  notion.} and split in \({\mathsf{Grp}}{\mathsf{Sch}}_{/R}\), then
\begin{align*}
\ker \qty{ H^1(R;G) \to H^1\qty{R \left[ { \scriptstyle f^{-1}} \right]; G}} = 1
.\end{align*}

\end{corollary}

\begin{remark}

This simplification comes from the injectivity of the following:
\begin{align*}
H^1( \widehat{R} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]; G) \hookrightarrow H^1(\widehat{K}  \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]; R)
.\end{align*}

In the limit, this says that many torsors are actually trivial. We find
that \(H^1_{\mathrm{zar}}(R; {\operatorname{SL}}_n) = 1\) and
\(H^1_{\mathrm{zar}}(R; E_8) = 1\).

\end{remark}

\hypertarget{curves-over-an-algebraically-closed-field}{%
\subsection{Curves Over an Algebraically Closed
Field}\label{curves-over-an-algebraically-closed-field}}

\begin{theorem}[?]

Let \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
and \(G \in {\mathsf{Alg}}{\mathsf{Grp}}_{_{/ {k}} }\) be semisimple and
\(C\) a smooth connected curve. Then
\(H^1_\mathrm{\operatorname{fppf}}(C; G) = 1\).

\end{theorem}

\begin{remark}

The main ingredient is Steinberg's theorem that \(H^1(K(C); G) = 1\). A
special case is \(\operatorname{PGL}_n\) and rephrases that central
simple algebras over \(k(C)\) are matrix algebras (using Tsen's
theorem). This also uses that \({\operatorname{Pic}}(C)\) is divisible,
which follows from the structure of \({\operatorname{Pic}}(C^c)\) for
\(C^c\) a smooth compactification.

\end{remark}

\begin{remark}

We have a degree map and moreover an exact sequence involving the
Jacobian (an abelian variety):
\begin{align*}
0 \to J_{C^c}(k) \to {\operatorname{Pic}}(C^c) \xrightarrow{\deg} {\mathbb{Z}}\to 0
.\end{align*}
If \(C\) is \(C^c\) minus finitely many points,
\({\operatorname{Pic}}(C^c)\twoheadrightarrow{\operatorname{Pic}}(C)\)
induces \(J_{C^c}(k) \twoheadrightarrow{\operatorname{Pic}}(C)\). We'll
sketch the proof first in the case \(G\) is simply connected. In this
case, given \(\gamma\in H^1(C; G)\), and according to Steinberg's
theorem there exists \(f\in k[C]\) with \(\gamma_{C^c} = ?\). In the
general case, we can take a simply connected cover \(f: \tilde G\to G\),
e.g.~\({\operatorname{SL}}_n\to \operatorname{PGL}_n\) or
\({\operatorname{Spin}}_n\to {\operatorname{SO}}_n\). Let \(\tilde T\)
be its maximal torus, then \(T\coloneqq\tilde T/\ker f\) is a maximal
torus in \(G\), so let \(B \subseteq G\) be a Borel containing \(T\).

\begin{claim}

\begin{align*}
H^1(C; B) \twoheadrightarrow H^1(C; G)
.\end{align*}

\end{claim}

Letting \(E\) be a \(C{\hbox{-}}\)torsor under \(G\), then the idea is
to introduce the twisted \(C{\hbox{-}}\)scheme \(Y\coloneqq E(G/B)\), a
projective variety of flags. By Steinberg's theorem,
\(Y(k(C)) \neq \emptyset\). Applying the valuative criterion of
properness shows that \(Y\) has a \(C{\hbox{-}}\)point, so \(E(G)\) has
a Borel subgroup scheme. By functoriality,
\begin{align*}
[E] \in \operatorname{im}(H^1(C; B) \to H^1(C; G))
.\end{align*}

We thus have \(B = U\rtimes T\) where \(U\) admits a
\(T{\hbox{-}}\)equivariant filtration with associated quotients
isomorphic to copies of \({\mathbb{G}}_a\), and we apply a dévissage
argument. Since \(\tilde T\to T\) is an isogeny (finite kernel) and
\({\operatorname{Pic}}(C)\) is a divisible group, a commutative diagram
shows surjectivity \(H^1(C; T)\twoheadrightarrow H^1(C; G)\) and thus
the latter is trivial.

The reductive case is similar, letting \(S = G/DG\) be the coradial
torus of \(G\) and showing \(H^1(C; G) \xrightarrow{\sim} H^1(C; S)\)
generalizing the bijection from yesterday:
\begin{align*}
H^1(C; \operatorname{GL}_r) \to H^1(C; {\mathbb{G}}_m) = {\operatorname{Pic}}(C)
.\end{align*}

\end{remark}

\hypertarget{affine-line}{%
\subsection{Affine Line}\label{affine-line}}

\begin{theorem}[?]

Let \(k \in \mathsf{Field}\) be not necessarily algebraically closed and
\(G\in{\mathsf{Grp}}{\mathsf{Sch}}_{_{/ {k}} }\) reductive. We have a
bijection
\begin{align*}
H^1(k; G) \xrightarrow{\sim} \ker\qty{ H^1(k[t]; G) \to H^1(k_s[t]; G)}
.\end{align*}
If \(k\) is perfect or \(\operatorname{ch}k = p\) where \(p\) is
``good'' for \(G\), we have \(H^1(k_s[t]; G) = 1\) and so
\(H^1(k; G) = H^1(k[t]; G)\) and we say \(G{\hbox{-}}\)torsors over
\(k[t]\) are \textbf{constant}.

\end{theorem}

\begin{remark}

This doesn't hold for \(G = \operatorname{PGL}_p\) over \(k(t)\) with
\(\operatorname{ch}k = p\) imperfect. Our next goal is to prove this
theorem -- a common ingredient to all proofs is the following theorem on
bundles over \({\mathbb{P}}^1\):

\end{remark}

\begin{theorem}[Grothendieck-Harder]

For \(G\) a reductive \(k{\hbox{-}}\)group, let \(S\) be a maximal
\(k{\hbox{-}}\)split torus of \(G\) and consider its constant associated
Weyl group
\begin{align*}
W_G(S) = N_G(S) / C_G(S)
.\end{align*}
Then there is a bijection
\begin{align*}
H^1_{\mathrm{zar}}({\mathbb{P}}^1_{_{/ {k}} }; S)/W_G(S) \xrightarrow{\sim} \ker\qty{H^1({\mathbb{P}}^1_{_{/ {k}} }; G) \xrightarrow{\operatorname{ev}_0}  H^1(k; C)}
.\end{align*}
In particular if a \(G{\hbox{-}}\)torsor over \(k[t]\) is trivial at
\(t=0\) and extends to a \(G{\hbox{-}}\)torsor over
\({\mathbb{P}}^1_{_{/ {k}} }\), then it is trivial.

\end{theorem}

\begin{remark}

Given a \(G{\hbox{-}}\)torsor over \(k[t]\), without loss of generality,
we can assume \(X\) is trivial on \(t=0\) -- the original method to
extend \(X\) to \({\mathbb{P}}^1\) is to use Bruhat-Tits theory. The
idea is to find an integer \(d\geq 1\) where \(\gamma_{k[t^d]}\) extends
to \({\mathbb{P}}^1\). The statement is local at \(\infty\), i.e.~it's
enough to find \(d\) where \(\gamma_{k((t^{-d}))}\) comes from
\(H^1(\cdots)\).

The following map is surjective:
\begin{align*}
H^1(k((t^{-1})); S)\to H^1(k((t^{-1})); G)
,\end{align*}
and we can write the absolute Galois group of
\(k{\left(\left( t^{-1} \right)\right) }\) as
\begin{align*}
\lim_n \mu_n(k_s) \rtimes G(k_s{}_{/ {k}} ) = I \rtimes G(k_s {}_{/ {k}} )
.\end{align*}
A restriction of a cocycle to the inertia group is a group morphism, so
factors through \(\mu_d(k_s)\) for some finite \(d\), which we can take
to be the order of \(S(k_s)\). We have some \(\gamma \in H^1(k[t]; G)\)
satisfying \(\gamma(0) = 1\), and a trick is to introduce a new
indeterminate \(u\) and to extend to \(F \coloneqq k(u)\).

The upshot is that
\begin{align*}
\operatorname{ff}\qty{ k(u, t, (ut)^{1\over d}) }
\cong
k(t,x)
.\end{align*}
by a \(k(t){\hbox{-}}\)linear isomorphism. The kernel is trivial by a
specialization argument, so \(\gamma\) is rationally trivial and extends
to infinity.

\end{remark}

\begin{remark}

Noting that
\({\mathbb{A}}^1 = {\mathbb{P}}^1 \setminus\left\{{{\operatorname{pt}}}\right\}\),
we have
\({\mathbb{G}}_m = {\mathbb{P}}^1\setminus\left\{{{\operatorname{pt}}_0, {\operatorname{pt}}_1}\right\}\),
which is much more difficult.

\end{remark}

\begin{theorem}[?]

Let \(G\) be a reductive \(k{\hbox{-}}\)group over
\(\operatorname{ch}k = 0\), then there is a bijection
\begin{align*}
H^1(k[t, t^{-1}]; G) \xrightarrow{\sim} H^1(k{\left(\left( t \right)\right)  }; G)
.\end{align*}

\end{theorem}

\begin{remark}

Surjectivity is easy, coming from reduction to a finite subgroup, and
injectivity is hard. A crucial step is to show existence of a maximal
torus for the relevant twisted group scheme, using again Bruhat-Tits
theory and now \emph{twin buildings}. So we have a good understanding of
\({\mathbb{G}}_m{\hbox{-}}\)torsors, and a next step would be
understanding \({\mathbb{P}}^n\) with more deleted points.

\end{remark}

\hypertarget{fruxe9duxe9ric-duxe9glise-talk-1-wednesday-july-14}{%
\section{Frédéric Déglise, Talk 1 (Wednesday, July
14)}\label{fruxe9duxe9ric-duxe9glise-talk-1-wednesday-july-14}}

\hypertarget{intro-3}{%
\subsection{Intro}\label{intro-3}}

\textbf{References:}

\begin{itemize}
\item
  \url{http://deglise.perso.math.cnrs.fr/docs/2021/PCMI1.pdf}
\item
  \href{https://www.ias.edu/sites/default/files/Deglise\%20Lecture\%201.pdf}{PCMI
  Notes 1}
\item
  \href{http://deglise.perso.math.cnrs.fr/docs/2021/PCMI2.pdf}{PCMI
  Notes 2}
\end{itemize}

\textbf{Abstract}:

\begin{quote}
Building on initial conjectures due to Beilinson, Voevodsky has
initiated a rich variety of ``motivic categories'', the universal one
being Morel-Voevodsky's homotopy category. This world, that is now
called ``motivic homotopy theory'', has produced a wide range of
results, settling older conjectures as well as opening new tracks to
follow.

This lecture series will aim at giving a survey of this world, from the
pure motivic origin, through the functoriality developments and then to
some of the exciting open questions.
\end{quote}

\begin{remark}

Recall the Euler product expansion for the zeta function. General
\(L{\hbox{-}}\)functions were studied around the 20s, followed by the
Weil conjectures in the 40s, and then étale \(\ell{\hbox{-}}\)adic
shaves by Grothendieck et al in the 60s. Letters from Grothendieck to
Serre describe the notion of \emph{weights} in relation to the Weil
conjectures, and served as an impetus in the early 70s for \emph{pure
motives}.

A second line of study considered number fields and class number
formulas, along with special values of \(L{\hbox{-}}\)functions, going
back to Dirichlet. Lichtenbaum related special values to
\(K{\hbox{-}}\)theory in the 70s, and this along with the theory of
perverse sheaves in the early 80s led to the Beilinson conjecture and
motivic complexes in the 90s.

As an aside, there is also a notion of \(p{\hbox{-}}\)adic
\(L{\hbox{-}}\)functions and corresponding \(p{\hbox{-}}\)adic motives.

\end{remark}

\begin{remark}

An outline for today:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Sheaves with transfers, which are modeled on étale homotopy sheaves
\item
  Homotopy sheaves over perfect fields
\item
  Motivic complexes
\end{enumerate}

\end{remark}

\begin{remark}

There are three main notions for étale sheaves:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Sheaves with transfers (see \emph{algebraic cycles}),
\item
  The (big) smooth Nisnevich site,
\item
  \({\mathbb{A}}^1{\hbox{-}}\)homotopy invariance.
\end{enumerate}

\end{remark}

\hypertarget{setting-up-sheaves-with-transfers}{%
\subsection{Setting up sheaves with
transfers}\label{setting-up-sheaves-with-transfers}}

We'll fix \(S\) a regular Noetherian scheme.

\hypertarget{finite-correspondences}{%
\subsubsection{Finite Correspondences}\label{finite-correspondences}}

\begin{definition}[Finite Correspondences]

For \(X,Y\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}\), a \textbf{finite
correspondence} \(\alpha\) from \(X\) to \(Y\) is a formal sum
\begin{align*}
\alpha = \sum_{i=1}^m m_i [Z_i]
&& \text{with }
Z_i \subseteq X \underset{\scriptscriptstyle {S} }{\times} Y \text{ closed, integral}
\end{align*}
with \(Z_i\to X\) finite and dominant over a connected component of
\(X\), i.e.~an algebraic cycle in the product. These form an abelian
group denoted \(c(X, Y) \in {\mathsf{Ab}}{\mathsf{Grp}}\), and can be
composed without imposing any equivalence relation on algebraic cycles.

We can thus define a closed symmetric monoidal (additive) category
enriched over abelian groups, the \textbf{category of finite
correspondences} over \(S\):
\begin{align*}
\mathsf{C} &\coloneqq{\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}  \\
\mathrm{Ob}(\mathsf{C}) &\coloneqq{\operatorname{Ob}}({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} ) \\
\mathsf{C}(X, Y) &\coloneqq c(X, Y)
.\end{align*}
where the monoidal structure is the cartesian product over \(S\) on
objects and on \(c(X, Y)\) is induced by the exterior product of
algebraic cycles.

\end{definition}

\begin{remark}

Writing
\(XYZ \coloneqq X { \underset{\scriptscriptstyle {S} }{\times} } Y { \underset{\scriptscriptstyle {S} }{\times} } Z\),
we have smooth projection maps
\begin{align*}
p: XYZ &\to XY \\
r: XYZ &\to XZ \\
q: XYZ &\to YZ 
.\end{align*}
Given cycles \(\alpha\in c(X, Y), \beta\in c(Y, Z)\), these pull back to
\(XYZ\) and intersect properly, with their intersection product given by
Serre's Tor formula.

\end{remark}

\begin{definition}[Graph]

Let \(Y \xrightarrow{f} X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}\), and
define the \textbf{graph of \(f\)} as the following pullback:

\begin{center}
\begin{tikzcd}
    {\Gamma_f} && {Y{ \underset{\scriptscriptstyle {S} }{\times} } X} \\
    \\
    X && {X{ \underset{\scriptscriptstyle {S} }{\times} } X}
    \arrow[from=1-1, to=3-1]
    \arrow[from=1-1, to=1-3]
    \arrow["\delta", from=3-1, to=3-3]
    \arrow["{f\times 1}", from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXEdhbW1hX2YiXSxbMiwwLCJZXFxmaWJlcnByb2R7U30gWCJdLFsyLDIsIlhcXGZpYmVycHJvZHtTfSBYIl0sWzAsMiwiWCJdLFswLDNdLFswLDFdLFszLDIsIlxcZGVsdGEiXSxbMSwyLCJmXFx0aW1lcyAxIl1d}{Link
to Diagram}
\end{quote}

Here \(\delta\) is the diagonal immersion of \(X_{/ {S}}\).

\end{definition}

\begin{remark}

Note that \(\Gamma_f \subseteq YX\) is a closed subscheme, and there is
an associated algebraic cycle
\begin{align*}
[\Gamma_f]_{XY} \in c(Y, X)
.\end{align*}

\end{remark}

\begin{definition}[Transpose]

Letting \({\varepsilon}: YX\to XY\) be the permutation of factors,
\({\varepsilon}_* [\Gamma_f] \in c(X, Y)\) is a finite correspondence
denoted \(f^t\), the \textbf{transpose} of \(f\).

\end{definition}

\begin{remark}

Several of the operations from the six functor formalism appear here:

\begin{itemize}
\tightlist
\item
  \textbf{Base change} can be defined for \(T \xrightarrow{f} S\) as
  \begin{align*}
  f^*: {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}  &\to {\mathsf{Cor}}{\mathsf{Sch}}_{/ {T}}  \\
  X & \mapsto X{ \underset{\scriptscriptstyle {S} }{\times} } T
  \end{align*}
  using \emph{pullback} for finite correspondences.
\item
  \textbf{Forgetting the base} is given by
  \begin{align*}
  p_\sharp: {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}  &\to {\mathsf{Cor}}{\mathsf{Sch}}_{/ {T}}  \\
  Y_{/ {T}}  &\mapsto Y_{/ {S}} 
  \end{align*}
  using \emph{direct images} for finite correspondences.
\end{itemize}

We now enlarge \({\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}\) to a larger
abelian category. This uses the fact that the Yoneda embedding will be a
fully faithful functor
\begin{align*}
X\mapsto c({-}, X) \coloneqq{\mathbb{Z}}^{\mathrm{tr}}_{/ {S}} (X)
\end{align*}
landing in a cocomplete abelian category extending the 6 functors.

\end{remark}

\hypertarget{presheaves-with-transfers}{%
\subsubsection{Presheaves with
transfers}\label{presheaves-with-transfers}}

\begin{definition}[Presheaf with transfers]

A \textbf{presheaf with transfers} \({\mathcal{F}}\) over \(S\) is an
additive functor
\begin{align*}
{\mathcal{F}}: {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} ^{\operatorname{op}}\to {\mathsf{Ab}}{\mathsf{Grp}}
.\end{align*}
We then define a \textbf{category of presheaves with transfers} over
\(S\):
\begin{align*}
\mathsf{C} &\coloneqq{\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}}  \\
{\operatorname{Ob}}(\mathsf{C}) &\coloneqq\text{Presheaves with transfers } {\mathcal{F}}\\
\mathsf{C}({\mathcal{F}}, {\mathcal{G}}) &\coloneqq\text{Natural transformations } \eta: {\mathcal{F}}\to{\mathcal{G}}
.\end{align*}

\end{definition}

\begin{remark}

Let \(f\in {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (Y, X)\). Note that by
contravariance of presheaves \({\mathcal{F}}\) we always get maps
\begin{align*}
{\mathcal{F}}(f) \in {\mathsf{Ab}}{\mathsf{Grp}}({\mathcal{F}}(X), {\mathcal{F}}(Y))
.\end{align*}
The data of a \emph{transfer} is the additional following operation on
\({\mathcal{F}}\), yielding a ``wrong way'' map:
\begin{align*}
f_* \coloneqq{\mathcal{F}}(f^t) \in {\mathsf{Ab}}{\mathsf{Grp}}( {\mathcal{F}}(Y), {\mathcal{F}}(X))
.\end{align*}

\end{remark}

\begin{example}[of presheaves with transfers]

\envlist

\begin{itemize}
\tightlist
\item
  \({\mathbb{G}}_m({-}): X_{/ {S}} \to {\mathcal{O}}_X(X)^{\times}\) has
  transfers over \(S\).
\item
  For \(A\in {\mathsf{Ab}}{\mathsf{Var}}_{/ {k}}\), the functor
  \(\mathop{\mathrm{Hom}}({-}, A)\) has transfers over \(k\).
\item
  \(H^n({-})\) has transfers over \(S\) for any mixed Weil cohomology
  theory.
\item
  \({\mathsf{K}}{\hbox{-}}\)theory does \textbf{not} form a presheaf
  with transfers. One instead needs \emph{unramified}
  \(K{\hbox{-}}\)theory.
\end{itemize}

\end{example}

\hypertarget{nisnevich-sheaves}{%
\subsubsection{Nisnevich Sheaves}\label{nisnevich-sheaves}}

\begin{definition}[Nisnevich Topology]

A \textbf{Nisnevich cover} of \(X\in {\mathsf{Sch}}\) is a family of
étale morphisms \(\left\{{ W_i \xrightarrow{p_i} X }\right\}_{i\in I}\)
where for \(x\in X\), \(p_i(w) = x\) for some \(w\in W_i\) inducing a
trivial residual extension \(\kappa(w) / \kappa(x)\).

\end{definition}

\begin{lemma}[Characterization of Nisnevich sheaves]

For
\({\mathcal{F}}: {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} ^{\operatorname{op}}\to {\mathsf{Ab}}{\mathsf{Grp}}\)
any abelian presheaf, \({\mathcal{F}}\) is a sheaf for the Nisnevich
topology iff \({\mathcal{F}}(\Delta)\) is a cartesian square for every
distinguished square \(\Delta\) of the following form:

\begin{center}
\begin{tikzcd}
    W && V \\
    & \Delta \\
    U && X
    \arrow["q", from=1-1, to=1-3]
    \arrow["k"', from=1-1, to=3-1]
    \arrow["j"', from=3-1, to=3-3]
    \arrow["p", from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJXIl0sWzAsMiwiVSJdLFsyLDAsIlYiXSxbMiwyLCJYIl0sWzEsMSwiXFxEZWx0YSJdLFswLDIsInEiXSxbMCwxLCJrIiwyXSxbMSwzLCJqIiwyXSxbMiwzLCJwIl1d}{Link
to Diagram}
\end{quote}

Here \(j\) is an open immersion, has reduced closed complement \(Z\),
\(p\) is étale, and \(p^{-1}(Z) \xrightarrow{\sim} Z\).

\end{lemma}

\hypertarget{sheaves-with-transfers}{%
\subsubsection{Sheaves with transfers}\label{sheaves-with-transfers}}

\begin{definition}[Sheaf with transfers]

There is a canonical embedding
\begin{align*}
\gamma: {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}  &\to {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}  \\
X &\mapsto X \\
(Y\to X) &\mapsto [\Gamma_f]_{XY} \in c(Y, X)
.\end{align*}
A \textbf{sheaf with transfers} is a presheaf
\({\mathcal{F}}\in {\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}}\)
such that
\begin{align*}
{\mathcal{F}}\circ \gamma: \in {\mathsf{Sh}}\qty{ {\mathsf{Sch}}^{{\mathrm{Nis}}}_{/ {S}} ,  {\mathsf{Ab}}{\mathsf{Grp}}}
,\end{align*}
i.e.~the composition \({\mathcal{F}}\circ \gamma\) is a sheaf on the
Nisnevich site of schemes (a \emph{Nisnevich sheaf}). These form a
category denoted \({\mathrm{tr}}{\mathsf{Sh}}_{/ {S}}\), and there is an
adjunction
\begin{align*}
\adjunction {\mathop{\mathrm{Forget}}} {a^{\mathrm{tr}}} {{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}}  } {{\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}}  }
\end{align*}
where
\({ \left.{{a^{\mathrm{tr}}({\mathcal{F}})}} \right|_{{{\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} }} } = \left( F\circ \gamma \right)^{\scriptscriptstyle \mathrm{sh}}\).

\end{definition}

\begin{remark}

The smooth site on \({\mathsf{Sch}}_{/ {S}}\) is big in the following
sense: to give a Nisnevich sheaf in this site is equivalent to an
assignment
\begin{align*}
{\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}  &\to {\mathsf{Sh}}({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} , {\mathsf{Ab}}{\mathsf{Grp}}) \\
X &\mapsto {\mathcal{F}}_X \\
(Y\xrightarrow{f} X) &\mapsto (f^*({\mathcal{F}}_Y) \xrightarrow{\tau_f} {\mathcal{F}}_X) 
.\end{align*}
Noting that \(\tau_f\) is not generally an isomorphism, it somehow
measures a defect of base change. In particular,
\({\mathsf{Sh}}({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} )\) is a much bigger
category than \({\mathsf{Sh}}({\mathsf{Sch}}^{\mathrm{Nis}}_{/ {S}} )\).

\end{remark}

\begin{example}[of sheaves with transfers]

\envlist

\begin{itemize}
\tightlist
\item
  For \(X_{/ {S}}\) smooth:
  \({\mathbb{Z}}_s^{\mathrm{tr}}(X) \coloneqq c_S({-}, X)\).
\item
  \({\mathbb{G}}_m({-}): X_{/ {S}} \to {\mathcal{O}}_X(X)^{\times}\).
\item
  For \(A\in {\mathsf{Ab}}{\mathsf{Var}}_{/ {k}}\),
  \(\mathop{\mathrm{Hom}}({-}, A)\).
\end{itemize}

As before, the last two examples don't form sheaves with transfers:

\begin{itemize}
\tightlist
\item
  \(H^n({-})\)
\item
  \({\mathsf{K}}{\hbox{-}}\)theory
\end{itemize}

\end{example}

\begin{remark}

We have \(f^*, p_\sharp, \otimes\) on
\({\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}}\), and these can be extended to
sheaves:

\begin{itemize}
\item
  \(f_*(F) \coloneqq F \circ f^*\), which yields a \textbf{base
  change/direct image} adjunction:
  \begin{align*}
  \adjunction{f^*}{f_*}{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} }{{\mathrm{tr}}{\mathsf{Sh}}_{/ {T}} }
  .\end{align*}
\item
  \(p:T\to S\) yields a \textbf{forget base/base change} adjunction:
  \begin{align*}
  \adjunction{p_\sharp}{p^*}{{\mathrm{tr}}{\mathsf{Sh}}_{/ {T}} }{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} }
  ,\end{align*}
  where for open immersions, \(p_\sharp\) is \(p_!\), the
  \textbf{exceptional direct image}
\item
  \({}_{h}\otimes\) on \({\mathrm{tr}}{\mathsf{Sh}}_{/ {S}}\) yields a
  \textbf{closed symmetric monoidal structure}
  \begin{align*}
  \adjunction{{-}\otimes^{\mathrm{tr}}{\mathcal{F}}}{\underline{\mathop{\mathrm{Hom}}}^{\mathrm{tr}}({\mathcal{F}}, {-})}{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} }{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} }
  ,\end{align*}
  where \(\underline{\mathop{\mathrm{Hom}}}^{\mathrm{tr}}\) is an
  internal hom.
\end{itemize}

\end{remark}

\hypertarget{homotopy-and-cohomology}{%
\subsection{Homotopy and Cohomology}\label{homotopy-and-cohomology}}

\hypertarget{mathbba1hbox-invariance-and-homotopic-morphisms}{%
\subsubsection{\texorpdfstring{\({\mathbb{A}}^1{\hbox{-}}\)invariance
and Homotopic
Morphisms}{\{\textbackslash mathbb\{A\}\}\^{}1\{\textbackslash hbox\{-\}\}invariance and Homotopic Morphisms}}\label{mathbba1hbox-invariance-and-homotopic-morphisms}}

\begin{definition}[$\AA^1$invariance and homotopy sheaves]

Let \({\mathcal{F}}\in {\mathrm{tr}}{\mathsf{Sh}}(S)\) and let
\(p:{\mathbb{A}}^1_{/ {X}} \to X\) be the canonical projection. We say
\({\mathcal{F}}\) is \textbf{\({\mathbb{A}}^1{\hbox{-}}\)invariant} or a
\textbf{homotopy sheaf} if for any
\(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}\), there is an induced
isomorphism
\begin{align*}
p^*: F(X) \xrightarrow{\sim} F({\mathbb{A}}^1_{/X})
.\end{align*}
These form a category denote \({\mathsf{HI}}^{\mathrm{tr}}(S)\).

\end{definition}

\begin{definition}[Homotopic morphisms]

Let \(\alpha, \beta \in {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (X, Y)\) be
two morphisms. We then say \textbf{\(\alpha\) is homotopic to \(\beta\)}
and write \(\alpha\sim\beta\) iff there exists a \(H\) satisfying the
following:
\begin{align*}
H &\in {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} ({\mathbb{A}}^1\times X, Y) \\
\alpha &= H \circ s_0 \\
\beta &= H \circ s_1
,\end{align*}
where \(s_0, s_1\) are the zero and unit sections of
\({\mathbb{A}}^{1}_{/X}\in \mathsf{Ring}{\mathsf{Sch}}_{/X}\). This
yields an equivalence relation, and we set
\begin{align*}
\pi_S(X, Y) \coloneqq{\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (X, Y)/\sim
.\end{align*}

\end{definition}

\begin{example}[of $\AA^1\dash$invariant sheaves]

The sheaves \({\mathbb{G}}_m({-})\) and
\(\mathop{\mathrm{Hom}}({-}, A)\) are
\({\mathbb{A}}^1{\hbox{-}}\)invariant.

\end{example}

\begin{theorem}[Suslin-Voevodsky]

Let \(S\in {\mathsf{Aff}}{\mathsf{Sch}}\) be regular and
\(C\in{\mathsf{Aff}}{\mathsf{Sch}}_{/ {S}}\) an affine curve admitting a
\textbf{good compactification} \(\tilde C\):

\begin{itemize}
\tightlist
\item
  \(\tilde C_{/ {S}}\) is proper and normal,
\item
  \(C\subseteq \tilde C\) is open/dense,
\item
  \(C_\infty \coloneqq\tilde C\setminus C\) admits an affine open
  neighborhood.
\end{itemize}

Then for any \(X\in {\mathsf{sm}}{\mathsf{Aff}}{\mathsf{Sch}}_{/ {S}}\),
there is a canonical isomorphism of groups:
\begin{align*}
\pi_S(X, C) &\xrightarrow{\sim} {\operatorname{Pic}}(X { \underset{\scriptscriptstyle {S} }{\times} } \tilde C { \underset{\scriptscriptstyle {S} }{\times} } C_\infty) \\
\alpha &\mapsto [{\mathcal{O}}(\alpha)]
,\end{align*}
where \({\mathcal{O}}(\alpha)\) is the line bundle associated to
\(\alpha\), viewed as a Cartier divisor in
\(X { \underset{\scriptscriptstyle {S} }{\times} } \tilde C\).

\end{theorem}

\hypertarget{cohomology-of-perfect-fields}{%
\subsubsection{Cohomology of Perfect
Fields}\label{cohomology-of-perfect-fields}}

\begin{definition}[Fiber functors]

Fix \(k \in \mathsf{Perf}\mathsf{Field}\), then a \textbf{function
field} \(E\) over \(k\) is a separable finitely generated field
extension \(E_{/ {k}}\). One can define the \textbf{fiber} of a homotopy
sheaf \(F\) at \(E_{/ {k}}\) as a filtered colimit over smooth finitely
generated sub \(k{\hbox{-}}\)algebras \(A\):
\begin{align*}
F(E_{/ {k}} ) \coloneqq\colim_{A_{/ {k}}  \leq E_{/ {k}} } F(\operatorname{Spec}A)
.\end{align*}
This yields a \textbf{fiber functor}: it is exact and commutes with
coproducts.

\end{definition}

\begin{remark}

We define the category
\({\mathsf{HI}}^{\mathrm{tr}}(S) \leq {\mathrm{tr}}{\mathsf{Sh}}(S)\) to
be the category of all homotopy sheaves, which is (Grothendieck) abelian
and bicomplete. The forgetful functor is exact, so there is an
adjunction
\begin{align*}
\adjunction
{h_0}
{\mathop{\mathrm{Forget}}}
{{\mathrm{tr}}{\mathsf{Sh}}_{/ {k}} }
{{\mathsf{HI}}^{\mathrm{tr}}_{/ {k}} }
.\end{align*}

\end{remark}

\begin{proposition}[Dévissage/purity]

If \(F\) is a homotopy sheaf and \(Z \xrightarrow{i} X\) is a
codimension 1 closed immersion in
\({\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\) with \(j: X\setminus Z\to X\) an
open immersion, then there is a SES of Nisnevich sheaves over
\(X_{\mathrm{Nis}}\):
\begin{align*}
0\to {\mathcal{F}}_X \to j_* {\mathcal{F}}_{X\setminus Z} \to i_* {\mathcal{F}}_{-1, Z} \to 0
.\end{align*}

\end{proposition}

\hypertarget{homotopy-invariance}{%
\subsubsection{Homotopy Invariance}\label{homotopy-invariance}}

\begin{theorem}[Main Theorem: homotopy invariance of homology]

If \(k\in \mathsf{Perf}\mathsf{Field}\) and
\({\mathcal{F}}\in {\mathsf{HI}}^{\mathrm{tr}}_{/ {k}}\), then for all
\(m\) and all \(X \in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\), there is an
isomorphism
\begin{align*}
p^*: H^n_{\mathrm{Nis}}(X; {\mathcal{F}}) \xrightarrow{\sim} H_{\mathrm{Nis}}^n({\mathbb{A}}^1_{/X}; {\mathcal{F}})
,\end{align*}
so the presheaf \(H^n_{\mathrm{Nis}}({-}, {\mathcal{F}})\) is homotopy
invariant.

\end{theorem}

\begin{corollary}[Purity theorem]

For \(Z\hookrightarrow X\) smooth closed of codimension \(m\), then
\begin{align*}
H_Z^n(X; {\mathcal{F}}) \xrightarrow{\sim} H_{\mathrm{Nis}}^{n-m}(Z; {\mathcal{F}}_{-m})
.\end{align*}
Here the LHS is Nisnevich cohomology with support.

\end{corollary}

\begin{corollary}[The Gersten resolution computes Nisnevich cohomology]

For \(X\) smooth, \({\mathcal{F}}_x\) is Cohen-Macaulay and there is a
Cousin complex \(C^*(X; {\mathcal{F}})\), also called the Gersten
complex of \({\mathcal{F}}\), and one can compute Nisnevich cohomology
as
\begin{align*}
H^n_{\mathrm{Nis}}(X; {\mathcal{F}}) \xrightarrow{\sim} H^n(C^*(X; {\mathcal{F}}))
.\end{align*}

\end{corollary}

\hypertarget{relation-to-chow}{%
\subsubsection{Relation to Chow}\label{relation-to-chow}}

\begin{theorem}[Nisnevich cohomology to Chow comparison]

Write
\({\mathbb{S}}^{n} \coloneqq{\mathbb{G}}_m{ {}^{ \scriptstyle {}_{h} {\otimes_{}^{n}} } }\),
then for a function field \(E_{/ {k}}\), there is an isomorphism of
sheaves
\begin{align*}
{\mathbb{S}}^n(E) \xrightarrow{\sim} \underline{ {\mathsf{K}}^{\scriptstyle\mathrm{M}} _n}(E) 
,\end{align*}
so this identifies with the \(n\)th unramified
\({\mathsf{K}}{\hbox{-}}\)theory of \(E\). Using the Gersten resolution
of \({\mathbb{S}}^{n}\), one obtains an isomorphism of groups
\begin{align*}
H_{\mathrm{Nis}}^n(X; {\mathbb{S}}^n) \xrightarrow{\sim} 
{\operatorname{CH}}^n(X)
,\end{align*}
the Chow group of codimension \(n\) cycles modulo rational equivalence.

\end{theorem}

\hypertarget{fruxe9duxe9ric-duxe9glise-talk-2-thursday-july-15}{%
\section{Frédéric Déglise, Talk 2 (Thursday, July
15)}\label{fruxe9duxe9ric-duxe9glise-talk-2-thursday-july-15}}

\hypertarget{intro-4}{%
\subsection{Intro}\label{intro-4}}

\begin{remark}

Recall the Beilinson conjectures (84/86), and Bloch's higher Chow groups
(86). Today we'll discuss the \({\mathbb{A}}^1{\hbox{-}}\)homotopy
category \({\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}\) and the
category of motives \({\mathsf{DM}}_{/ {S}}\). We'll be working
\({\mathsf{Top}}_*\), the infinity category of pointed spaces, and
\(\mathbf{D} {{\mathsf{Ab}}}\), the (infinity) derived category of
abelian groups.

\end{remark}

\hypertarget{the-homotopy-category}{%
\subsection{The homotopy category}\label{the-homotopy-category}}

\begin{definition}[The homotopy category]

Consider infinity functors
\begin{align*}
F: {\mathsf{sm}}{\mathsf{Sch}}^{\operatorname{op}}_{/ {S}} \to {\mathsf{Top}}_*
\end{align*}
and define
\begin{align*}
F(X, Z) \coloneqq{\operatorname{hofib}}( F(X) \to F(X\setminus Z))
.\end{align*}
Then \textbf{the (pointed) \({\mathbb{A}}^1\) homotopy category of
\(S\)}, denoted
\({\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}\), consists of
such functors \(F\) that satisfy

\begin{itemize}
\item
  \textbf{Excision}: for all \((Y, T) \to (X, Z)\) excisive, there is a
  weak equivalence
  \begin{align*}
  F(Y, T) \underset{\scriptscriptstyle W}{\rightarrow}F(X, Z)
  .\end{align*}
\item
  \textbf{Homotopy invariance}: The canonical projection
  \({\mathbb{A}}^1_{/ {X}}\) induces a weak equivalence
  \begin{align*}
  F(X) \underset{\scriptscriptstyle W}{\rightarrow}F({\mathbb{A}}^1_{/ {X}} )
  .\end{align*}
\end{itemize}

This category admits a monoidal structure, which we'll denote by the
smash product \(X\wedge Y\).

\end{definition}

\begin{remark}

The excision axiom can be replaced by the following condition: for
distinguished squares \(\Delta\), the image \(F(\Delta)\) is homotopy
cartesian:

\begin{center}
\begin{tikzcd}
    {W_+} && {V_+} \\
    & {} \\
    {U_+} && {X_+}
    \arrow[from=1-1, to=1-3]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=1-1, to=3-1]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJXXysiXSxbMiwwLCJWXysiXSxbMCwyLCJVXysiXSxbMiwyLCJYXysiXSxbMSwxXSxbMCwxXSxbMiwzXSxbMSwzXSxbMCwyXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=}{Link
to Diagram}
\end{quote}

We can similarly ask for (infinity additive) functors
\(K:{\mathsf{sm}}{\mathsf{Sch}}^{\operatorname{op}}_{/ {S}} \to \mathbf{D} {{\mathsf{Ab}}}\)
satisfying these properties.

\end{remark}

\begin{remark}

We can use infinity categorical localization theory to build this
category. For a scheme \(S\), view a \emph{pointed space over \(S\)} as
a presheaf valued in pointed simplicial sets, viewed as an infinity
category. We can then construct
\begin{align*}
{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}  &= 
 \underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} , \mathsf{sSet}_*) \left[ { \scriptstyle W^{-1}} \right] \coloneqq\mathsf{C} \left[ { \scriptstyle W^{-1}} \right] 
\\ \\
W &\coloneqq\left\{{
{\mathbb{Z}}_S^*({\mathbb{A}}^1_{/ {X}} ) \to {\mathbb{Z}}_S^*(X) {~\mathrel{\Big|}~}X\in {\operatorname{Ob}}(\mathsf{C})
}\right\}
.\end{align*}

\end{remark}

\begin{remark}

One can similarly do this for
\({\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} , \mathbf{D} { {\mathsf{Ab}}} ) = \mathbf{D} { {\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}} }\).
Effective motives \({\mathsf{DM}}^{\mathsf{eff}}_{/ {S}}\) can be
constructed by replacing presheaves with
\(\mathbf{D} {{\mathsf{Sh}}^{\mathrm{tr}}_{/ {S}} }\) and localizing at
\({\mathbb{Z}}_S^{\mathrm{tr}}({\mathbb{A}}^1_{/ {X}} ) \to {\mathbb{Z}}_S^{{\mathrm{tr}}}(X)\).

\end{remark}

\hypertarget{mathbba1hbox-locality}{%
\subsection{\texorpdfstring{\({\mathbb{A}}^1{\hbox{-}}\)locality}{\{\textbackslash mathbb\{A\}\}\^{}1\{\textbackslash hbox\{-\}\}locality}}\label{mathbba1hbox-locality}}

\begin{definition}[$\AA^1\dash$local spaces]

\textbf{\({\mathbb{A}}^1{\hbox{-}}\)local spaces} are those
\(S{\hbox{-}}\)spaces for which the realization induces a weak
equivalence on mapping spaces:
\begin{align*}
{\left\lvert {{\mathbb{A}}^1_{/ {X}}  } \right\rvert} _+ 
\to X_+ 
\leadsto 
\mathop{\mathrm{Hom}}(X_+, Y ) 
\underset{\scriptscriptstyle W}{\rightarrow}
\mathop{\mathrm{Hom}}(\qty{ {\mathbb{A}}^1_{/ {X}}  } _+, Y) \quad \forall Y\in {\operatorname{Ob}}(\mathsf{C})
.\end{align*}

\end{definition}

\begin{remark}

Fix \(k\in\mathsf{Perf}\mathsf{Field}\) and consider complexes of
sheaves \(K \in\mathsf{Ch}\qty{ {\mathsf{Sh}}^{\mathrm{tr}}_{/ {S}} }\).
We can define cohomology sheaves \(H^*(K)\) by taking kernels mod images
in \(\underset{ \mathsf{pre} } {\mathsf{Sh} }^{\mathrm{tr}}_{/ {S}}\)
and Nisnevich-sheafifying to get a sheaf
\begin{align*}
\underline{H}^i(K) \coloneqq \left( H^i(K) \right)^{\scriptscriptstyle \mathrm{sh}}  \in {\mathsf{Sh}}^{\mathrm{tr}}_{/ {S}} 
.\end{align*}
This gives a way to take cohomology of complexes of sheaves with
transfers.

\end{remark}

\begin{theorem}[Characterization of $\AA^1\dash$local complexes of sheaves]

\(K\) is \({\mathbb{A}}^1{\hbox{-}}\)local iff for all
\(\underline{H^n}(K)\) is \({\mathbb{A}}^1{\hbox{-}}\)local in
\({\mathsf{HI}}^{\mathrm{tr}}_{/ {k}}\) for all \(n\)

\end{theorem}

\begin{definition}[Suslin Complex]

Define \textbf{standard cosimplicial scheme} as
\begin{align*}
\Delta^n\coloneqq\operatorname{Spec}\qty{k[x_0, \cdots, x_n] \over \left\langle{ \sum x_i  }\right\rangle } \in {\mathsf{Sch}}_{/ {k}} 
\end{align*}
and for \(K\in \mathsf{Ch}\qty{ {\mathrm{tr}}{\mathsf{Sh}}_{/ {k}} }\) a
complex of sheaves with transfers, the \textbf{Suslin singular complex}
is the complex of sheaves defined as
\begin{align*}
\underline{C_{*}^S}(K), && {\mathsf{\Gamma}\qty{X_{/ {S}} } } \coloneqq{ \operatorname{Tot} }^{\Pi}K(\Delta^\bullet { \underset{\scriptscriptstyle {k} }{\times} } X)
.\end{align*}
for \(X\in{ {\mathsf{sm}}{\mathsf{Sch}}}_{/ {S}}\).

\end{definition}

\begin{corollary}[The Suslin complex is $\AA^1\dash$local]

The Suslin singular complex \(\underline{C_*^S}(K)\) is
\({\mathbb{A}}^1{\hbox{-}}\)local, and the functor
\(\underline{C_*^S}({-})\) is an isomorphism in \({\mathsf{DM}}\)?

\end{corollary}

\hypertarget{motives}{%
\subsection{Motives}\label{motives}}

\begin{definition}[Motives]

The \textbf{homological motive} of a smooth scheme
\(X\in{ {\mathsf{sm}}{\mathsf{Sch}}}_{/ {k}}\) is
\begin{align*}
M(X) \coloneqq C_*(S){\mathbb{Z}}^{\mathrm{tr}}_{/ {k}} (X)
.\end{align*}

\end{definition}

\begin{definition}[Tate twist]

The \textbf{Tate twist} is defined as
\begin{align*}
{\mathbb{Z}}(1) \coloneqq\operatorname{coker}\qty{ M\left\{{1}\right\} \to M({\mathbb{G}}_m)}[-1] 
.\end{align*}

\todo[inline]{What is $\left\{{1}\right\}$? Maybe $\operatorname{Spec}k$...}

\end{definition}

\begin{example}[of identifying a Tate twist]

\({\mathbb{Z}}(1) = {\mathbb{G}}_m[-1]\in [0, 1]\) is supported in
homotopy degree 0 and 1 (and in fact just in degree 1), and generally
\({\mathbb{Z}}(n) = {\mathbb{Z}}(1){ {}^{ \scriptstyle\otimes_{k}^{n} } } \in (-\infty, n]\)
is supported in degree at most \(n\).

\end{example}

\begin{conjecture}[Reinforced Beilinson-Soulé]

For all \(n>0\), \({\mathbb{Z}}(n) \in [1, n]\), so it is in fact only
supported in positive degrees. Moreover, for
\(E_{/ {k}} \in{\mathsf{fn}}\mathsf{Field}\),
\begin{align*}
H^{i> n }\qty{ C_E(\Delta_E^*, {\mathbb{G}}_m^n)_{\mathbb{Q}}} = 0
.\end{align*}
By Bloch-Kato, the integral and rational cases are equivalent.

\end{conjecture}

\hypertarget{motivic-cohomology-1}{%
\subsection{Motivic Cohomology}\label{motivic-cohomology-1}}

\begin{definition}[Motivic cohomology]

For \(X\in{ {\mathsf{sm}}{\mathsf{Sch}}}_{/ {k}}\), the \textbf{motivic
cohomology} is given by
\begin{align*}
H_{ \mathrm{mot}} ^{n, i}(X) \coloneqq H^n_{\mathrm{Nis}}(X; {\mathbb{Z}}(i))
.\end{align*}
The grading \(n\) is the \textbf{degree}, and \(i\) is the
\textbf{twist}.

\end{definition}

\begin{remark}

Let
\({\mathbb{Z}}(m)\in \mathbf{D} {{\mathsf{Sh}}^{\mathrm{tr}}_{/ {k}} }\),
then for \(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\) we have
\begin{align*}
H^{n, i}_M(X) = \mathop{\mathrm{Hom}}_{{\mathsf{DM}}^\mathsf{eff}}(M(X), {\mathbb{Z}}(i)[n] ) 
.\end{align*}
Taking the sheaf defined in top diagonal bidegree, this can be
identified with unramified Milnor \({\mathsf{K}}{\hbox{-}}\)theory:
\begin{align*}
\operatorname{\mathcal{H}}^n({-}; {\mathbb{Z}}(n)) = \operatorname{\mathcal{K^M}}_n({-})
.\end{align*}

\end{remark}

\hypertarget{stable-six-functors}{%
\subsection{Stable Six Functors}\label{stable-six-functors}}

\begin{remark}

Let
\(X: { {\mathsf{sm}}{\mathsf{Sch}}}^{\operatorname{op}}_{/ {S}} \to {\mathsf{Top}}_*\),
which is a ``space'' in an infinity categorical sense, and consider
\(f:T\to S\) a morphisms of schemes. We can form
\(f^*: { {\mathsf{sm}}{\mathsf{Sch}}}_{/ {S}} \to { {\mathsf{sm}}{\mathsf{Sch}}}_{/ {T}}\)
which induces an adjunction
\begin{align*}
\adjunction {f^*} {f_*} 
{{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} }  
{{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {T}} }
.\end{align*}
For \(p:T\to S\) smooth, we can define \(p_\sharp\) and \(p^*\)
similarly, yielding
\begin{align*}
\adjunction {p_\sharp} {p^*} 
{{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} }  
{{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {T}} }
.\end{align*}
There is also a stable lift of the tensor product to a smash product
\({-}\wedge{-}\), yielding
\begin{align*}
\adjunction {{-}\wedge{-}} {\underline{\mathop{\mathrm{Hom}}}({-}, {-})} 
{{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} }  
{{\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} }
.\end{align*}

\todo[inline]{Not precise, need to apply a space as an argument...?}

There are also formulas for things like \(f^*(K\wedge X_+)\), as well as
(smooth) base change and projection.

\end{remark}

\begin{theorem}[Morel-Voevodsky Localization]

Let \(i: Z\hookrightarrow S\) be closed and \(U\coloneqq S\setminus Z\)
with \(j: U \underset{\scriptscriptstyle O}{\hookrightarrow}S\) an open
immersion. Then for all
\(X\in {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}\), there is
a homotopy cofiber sequence
\begin{align*}
j_\sharp j^*(X) \to X\to i_* i^* X
,\end{align*}
where the maps are given by units/counits of the corresponding
adjunctions.

\end{theorem}

\begin{remark}

This can be restated as a geometric version of
\({\mathbb{A}}^1{\hbox{-}}\)homotopy equivalence: that there is a weak
equivalence
\begin{align*}
{X \over X \setminus(X{ \underset{\scriptscriptstyle {S} }{\times} } Z)} \underset{\scriptscriptstyle W}{\rightarrow}i_*((X_Z)_+)
.\end{align*}
We don't have the 6 functor formalism unstably.

\end{remark}

\hypertarget{stabilization}{%
\subsection{Stabilization}\label{stabilization}}

\begin{remark}

One can take spheres in
\({\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}\) to be the
pointed space
\begin{align*}
({\mathbb{P}}^1, \infty ) \simeq S^1 \wedge({\mathbb{G}}_m, 1)
.\end{align*}
This yields a definition of loop spaces:
\begin{align*}
{\Omega}_{{\mathbb{P}}^1}({-}) \coloneqq{\mathbf{R}}\underline{\mathop{\mathrm{Hom}}}({\mathbb{P}}^1, {-})
,\end{align*}
where one needs to derive this construction.

\end{remark}

\begin{definition}[Stable homotopy category]

The \textbf{stable homotopy category \({\mathsf{SH}}_{/ {S}}\)} is
defined as the limit
\begin{align*}
\cdots 
\xrightarrow{{\Omega}_{{\mathbb{P}}^1}} {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} 
\xrightarrow{{\Omega}_{{\mathbb{P}}^1}} {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} 
\xrightarrow{{\Omega}_{{\mathbb{P}}^1}} {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}} 
,\end{align*}
which is a construction that works for presentable monoidal infinity
categories.

\end{definition}

\begin{remark}

This makes \({\mathbb{P}}^1\) a monoidally invertible object, and it
turns out to invert \({\mathbb{G}}_m\) and the classical sphere \(S^1\).
This is because if we define
\({\mathbb{S}}^{n, m} \coloneqq S^n \wedge{\mathbb{G}}_m^m\), we have
\begin{align*}
{\mathbb{P}}^1 &\simeq{\mathbb{S}}^{1, 1} && \coloneqq S^1 \wedge{\mathbb{G}}_m \\
{\mathbb{A}}^n\setminus\left\{{0}\right\}&\simeq{\mathbb{S}}^{n-1, n} && \coloneqq S^n \wedge{\mathbb{G}}_m{ {}^{ \scriptstyle\otimes_{k}^{n} }  }
.\end{align*}

\end{remark}

\begin{remark}

A concrete model is given by sequences of objects, called
\textbf{\({\mathbb{P}}^1{\hbox{-}}\)spectra}. These are sequences of
pointed spaces \(X\coloneqq(X_m)\) with
\({\mathbb{A}}^1{\hbox{-}}\)homotopy equivalences
\begin{align*}
X_m \underset{\scriptscriptstyle W}{\rightarrow}{\Omega}_{{\mathbb{P}}^1}(X_{m+1})
.\end{align*}
This is somehow related to
\({\mathbb{P}}^1\wedge X_m \xrightarrow{\sigma_m} X_{m+1}\).
\({\mathsf{SH}}_{/ {S}}\) satisfies the following universal property: it
is the universal presentable monoidal infinity category receiving a
functor
\begin{align*}
{\Sigma}^\infty : {\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}  \to {\mathsf{SH}}_{/ {S}} 
\end{align*}
such that \({\mathbb{P}}^1\wedge({-})\) is invertible. It turns out that
the category \({\mathsf{SH}}_{/ {S}}\) admits a diagram relating it to
all of the categories that have appeared thus far.

\end{remark}

\begin{theorem}[?-Voevodsky]

For \(f:T\to S\) a morphism of schemes, separated of finite type, there
is a triangulated adjunction
\begin{align*}
\adjunction{f_!}{f^!}{{\mathsf{SH}}_{/ {S}} } {{\mathsf{SH}}_{/ {T}}  }
\end{align*}
such that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(f_!\) is compatible with composition.
\item
  If \(f\) is proper then there is am isomorphism
  \(\eta: f_! \xrightarrow{\sim} f_*\).
\item
  If \(f\) is smooth, then
  \begin{align*}
  f_! = f_\sharp( \mathop{\mathrm{Th}}(T_f) \otimes({-}))
  \end{align*}
  where \(T_f\) is the tangent bundle and
  \begin{align*}
  \mathop{\mathrm{Th}}(T_f) \coloneqq{\Sigma}^\infty(T_f/T_f {}^{ \vee })
  \end{align*}
  is its associated Thom space.
\end{enumerate}

Moreover \(\mathop{\mathrm{Th}}(T_f)\) is tensor-invertible in
\({\mathsf{SH}}_{/ {S}}\) with inverse \(\mathop{\mathrm{Th}}(-T_f)\).

\end{theorem}

\begin{remark}

There is a base change formula, and \(p^* f_! \cong g_! q^*\) for
cartesian squares:

\begin{center}
\begin{tikzcd}
    Y && T \\
    \\
    X && S
    \arrow["g", from=1-1, to=1-3]
    \arrow["f", from=3-1, to=3-3]
    \arrow["q"', from=1-1, to=3-1]
    \arrow["p", from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJZIl0sWzIsMCwiVCJdLFswLDIsIlgiXSxbMiwyLCJTIl0sWzAsMSwiZyJdLFsyLDMsImYiXSxbMCwyLCJxIiwyXSxbMSwzLCJwIl1d}{Link
to Diagram}
\end{quote}

There is a projection formula
\begin{align*}
f_!(C \otimes f^*(F)) \cong f_!(C) \otimes F
.\end{align*}
Moreover, \({\mathsf{SH}}({-})\) satisfies a generalized Beilinson
formalism.

\end{remark}

\hypertarget{rational-homotopy}{%
\subsection{Rational Homotopy}\label{rational-homotopy}}

\begin{remark}

\({\mathsf{SH}}_{/ {S}}\) is triangulated, and there are several ways to
construct a triangulated rationalization
\({\mathsf{SH}}_{/ {S}} \otimes{\mathbb{Q}}\). This decomposes as
\begin{align*}
{\mathsf{SH}}_{/ {S}}  \otimes{\mathbb{Q}}\xrightarrow{\sim} \qty{{\mathsf{SH}}_{/ {S}} }_{{\mathbb{Q}}^+} \times \qty{{\mathsf{SH}}_{/ {S}} }_{{\mathbb{Q}}^-}
.\end{align*}

\begin{itemize}
\tightlist
\item
  The plus part is characterized by the algebraic Hopf map \(\eta\)
  acting by zero, \({\varepsilon}= -1\)
\item
  The minus by \(\eta\) being invertible and \({\varepsilon}= +1\)
\end{itemize}

For \(S\) regular, the plus part is equivalent to
\(\qty{ {\mathsf{DM}}_{/ {S}} }_{\mathbb{Q}}\). Writing
\(S^0 \coloneqq S\otimes_{\mathbb{Z}}{\mathbb{Q}}\), the minus part is
equivalent to the Witt sheaf
\(\operatorname{\mathcal{W}}^{{\mathbb{Q}}}_{S^0}\), which is connected
to quadratic forms. Reindexing and setting
\(\tilde {\mathbb{S}}^{n, i} \coloneqq S^{n-i} \wedge{\mathbb{G}}_m{ {}^{ \scriptstyle\otimes_{k}^{i} } }\),
one can define cohomotopy groups
\begin{align*}
\qty{\pi^{n, i}_{/ {S}} }_{\mathbb{Q}}
&\coloneqq
[ {\mathbb{S}}, S^{n-i} \wedge{\mathbb{G}}_m^i ]
_{\qty{{\mathsf{SH}}_{/ {S}} } _{\mathbb{Q}}} \\
& =[ {\mathbb{S}}, \tilde {\mathbb{S}}^{n, i}]
_{\qty{{\mathsf{SH}}_{/ {S}} } _{\mathbb{Q}}} \\
&= [\one, \one(s)[i] ] \\
& \xrightarrow{\sim} 
{\mathsf{gr}\,}_\gamma^i \qty{ (K_{2i-n}) _{/ {S}} }_{\mathbb{Q}}\oplus H^{n-i}_{\mathrm{Nis}}(S^0; \operatorname{\mathcal{W}} )
,\end{align*}
where \({\mathsf{gr}\,}\) is a grading.

For \(E\in\mathsf{Field}\), this yields
\begin{align*}
\pi^{n, i}(E)_{\mathbb{Q}}= H_{ \mathrm{mot}} (E)_{\mathbb{Q}}\oplus W(E)_{\mathbb{Q}}
.\end{align*}

\end{remark}

\begin{remark}

There is a Grothendieck-Verdier duality: for \(f:X\to S\) smooth finite
type with \(S\) regular, then
\(f^!(\one_S) \simeq\mathop{\mathrm{Th}}(Lf)\). If \({\mathbb{E}}\) is a
compact (constructible) object of \({\mathsf{SH}}_{/ {S}}\) the
\({\mathbb{E}} {}^{ \vee }= \underline{\mathop{\mathrm{Hom}}}({\mathbb{E}}, D_*)\)
and there is an isomorphism
\({\mathbb{E}}\to ({\mathbb{E}} {}^{ \vee }) {}^{ \vee }\).

\end{remark}

\hypertarget{kirsten-wickelgren-talk-1-wednesday-july-14}{%
\section{Kirsten Wickelgren, Talk 1 (Wednesday, July
14)}\label{kirsten-wickelgren-talk-1-wednesday-july-14}}

\hypertarget{intro-5}{%
\subsection{Intro}\label{intro-5}}

\textbf{Abstract}:

\begin{quote}
Morel and Voevodsky's \({\mathbb{A}}^1\) homotopy theory imports tools
from algebraic topology into the study of schemes, or in other words,
into the study of the solutions to polynomial equations. This theory
produces greater understanding of arithmetic and geometric aspects these
solutions. We will introduce some of this theory using as a guide
questions such as ``How many lines meet 4 lines in 3-space?''
\end{quote}

\textbf{References/Background}:

\begin{itemize}
\tightlist
\item
  Some algebraic topology or algebraic geometry, for example as
  described in Hatcher's and Hartshorne's books
\item
  \href{https://www.ias.edu/sites/default/files/L1\%20-\%20Intro\%20to\%20A1-homotopy\%20theory\%20using\%20enumerative\%20examples\%20-\%20Jul\%2013\%2C\%202021_1.pdf}{Lecture
  Notes 1}
\item
  \href{https://www.ias.edu/sites/default/files/L2\%20-\%20Intro\%20to\%20A1-homotopy\%20using\%20enumerative\%20examples\%20-\%20Jul\%2015\%2C\%202021.pdf}{Lecture
  Notes 2}
\item
  \href{https://www.ias.edu/sites/default/files/Pauli-Exercises_PCMI.pdf}{Exercises}
\end{itemize}

\begin{remark}

Enumerative geometry counts algebro-geometric objects over
\({\mathbb{C}}\). Example: how many lines meet 4 generic lines in
\({\mathbb{P}}^3\)? The answer is 2, and our goal is to record this kind
of arithmetic information about geometric objects over a field \(k\)
whose intersections are fixed over
\(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) but not
necessarily \(k\) itself. Our main tool will be
\({\mathbb{A}}^1{\hbox{-}}\)homotopy theory, due to Morel-Voevodsky.

\end{remark}

\hypertarget{classical-theory}{%
\subsection{Classical Theory}\label{classical-theory}}

\begin{remark}

First some classical homotopy theory. The sphere can be defined as
\begin{align*}
S^n \coloneqq\left\{{{ {x}_1, {x}_2, \cdots, {x}_{n}} {~\mathrel{\Big|}~}\sum x_i^2 = 1}\right\} \simeq{\mathbb{P}}^n({\mathbb{R}}) / {\mathbb{P}}^{n-1}({\mathbb{R}})
,\end{align*}
and we have a degree map \([S^n, S^n] \to {\mathbb{Z}}\). Given any
\(f\in {\mathsf{Top}}(S^n, S^n)\) and \(p\in S^n\), we can write
\(f^{-1}(p) = \left\{{{ {q}_1, {q}_2, \cdots, {q}_{N}}}\right\}\) and
compute \(\deg f = \sum_{i=1}^N \deg_{q_i} f\) in terms of \textbf{local
degrees}. Letting \(V\) be a ball containing \(p\), we have
\(F^{-1}(V) \supseteq U \ni q_i\) another ball such that
\(U \cap f^{-1}(p) = q_i\). Then
\(U/{{\partial}}U \simeq S^n \simeq V/{{\partial}}V\), so we can define
a map
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu: {U \over U\setminus\left\{{q_i}\right\}} \to {V \over V\setminus\left\{{p}\right\}}
\end{align*}
and define
\(\deg_{q_i} f \coloneqq\deg \mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu\).

Letting \({\left[ {{ {x}_1, {x}_2, \cdots, {x}_{n}}} \right]}\) be
oriented coordinates about \(q_i\) and
\({\left[ {{ {y}_1, {y}_2, \cdots, {y}_{n}}} \right]}\) about \(p\),
then
\(f = {\left[ {{ {f}_1, {f}_2, \cdots, {f}_{n}}} \right]}: {\mathbb{R}}^n\to {\mathbb{R}}^n\)
and we can consider
\(J_f \coloneqq\operatorname{det}\qty{{\partial}f_i \over {\partial}x_i}\).
There is then a formula
\begin{align*}
\deg_{q_i}(f) = 
\begin{cases}
+1 & J_f(q_i) > 0 
\\
-1 & J_f(q_i) < 0.
\end{cases}
,\end{align*}
and for all \(q_i\) we have \(\deg f = \# f^{-1}({\operatorname{pt}})\),
i.e.~the number of solutions to the polynomial system
\(\left\{{f_1 = f_2 = \cdots f_n = 0}\right\}\).

\end{remark}

\begin{example}[?]

If \(f\in {\mathbb{C}}[x]\) of degree \(n\), we can regard \(f\) as a
function
\(f: {\mathbb{P}}^1({\mathbb{C}})\to {\mathbb{P}}^1({\mathbb{C}})\) and
by the fundamental theorem of algebra,
\begin{align*}
\deg f = n = \# \left\{{f=0}\right\}
.\end{align*}

\end{example}

\begin{remark}

We can similarly count solutions to \(f=0\) when \(f\) is a section of a
rank \(n\) vector bundle

\begin{center}
\begin{tikzcd}
V
\ar[d, "p"] 
\\
X 
\ar[u, bend left, dotted, "f"]
\end{tikzcd}
\end{center}

This count can be computed using the Euler class:
\begin{align*}
e(V) = e(V, f) = \sum_{q_i\in \left\{{f=0}\right\}} \deg q_i f
.\end{align*}

\end{remark}

\begin{example}[?]

Let \(X \coloneqq{\operatorname{Gr}}(1, 3)_{/{\mathbb{C}}}\), the
Grassmannian parameterizing dimension 2 subspaces
\(W \subseteq {\mathbb{C}}^4\), or equivalently lines in
\({\mathbb{P}}W \subseteq {\mathbb{P}}({\mathbb{C}}^4) \cong {\mathbb{P}}^3({\mathbb{C}})\),
where \({\mathbb{P}}W\) is defined as \(W\setminus\left\{{0}\right\}\)
where \(\lambda w\sim w\) for any \(\lambda \in {\mathbb{C}}^{\times}\).
The tautological is a rank 2 bundle:

\begin{center}
\begin{tikzcd}
S_{[{\mathbb{P}}W]} = W
  \ar[r] 
& 
S
  \ar[d] 
\\
& 
{\operatorname{Gr}}(1, 3)_{/ {{\mathbb{C}}}} 
\end{tikzcd}
\end{center}

Let \(L_1, \cdots, L_4\) be four lines in \({\mathbb{P}}^3\), then
\(\left\{{\text{lines intersecting all } L_i}\right\} = \left\{{f=0}\right\}\)
where \(f\) is a section (depending on the \(L_i\)) of the bundle

\begin{center}
\begin{tikzcd}
{\mathcal{E}}\coloneqq\qty{\bigwedge\nolimits^2 S^*}^{\oplus 4}\
\ar[d] 
\\
{\operatorname{Gr}}(1, 3)
\end{tikzcd}
\end{center}

and the Euler number of this bundle counts the number of such
intersections. In particular, \(e({\mathcal{E}})\) is independent of the
choice of lines and section, provided they're sufficiently generic (so
the \(L_i\) do not pairwise intersect). Using the splitting principle
and knowledge of \(H^*({\operatorname{Gr}})\), one can compute
\(e({\mathcal{E}}) = 2\).

\end{example}

\hypertarget{over-arbitrary-fields-grothendieck-witt}{%
\subsection{Over arbitrary fields:
Grothendieck-Witt}\label{over-arbitrary-fields-grothendieck-witt}}

\begin{remark}

We'd like to do this over arbitrary fields \(k\). Lannes and Morel
defined degrees for rational maps
\(f:{\mathbb{P}}^1 \to {\mathbb{P}}^1\). Above we only remembered the
sign of \(J_f\), and here we'll allow remembering more: \(\deg f\) will
be valued in \({\operatorname{GW}}(k)\). We can realize
\({\operatorname{GW}}(k)\) as the group completion of the semiring of
nondegenerate symmetric bilinear forms under \(\perp, \otimes_k\), where
we complete with respect to \(\perp\). It is related to the Witt group
by
\begin{align*}
W(K) { { \, \xrightarrow{\sim}\, }}{{\operatorname{GW}}(k) \over {\mathbb{Z}}\left\langle{q_{ {\operatorname{hyp}}} }\right\rangle } \coloneqq{{\operatorname{GW}}(k) \over {\mathbb{Z}} { \left[ {\left\langle{1}\right\rangle + \left\langle{-1}\right\rangle} \right] } }
.\end{align*}
There is a \textbf{rank map}
\begin{align*}
\operatorname{rank}: {\operatorname{GW}}(k) &\to {\mathbb{Z}}\\
q: (V{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \to K) &\mapsto \dim_k V
,\end{align*}
which can be realized by a pullback

\begin{center}
\begin{tikzcd}
    {{\operatorname{GW}}(k)} && {W(k)} & {} \\
    \\
    {\mathbb{Z}}&& {{\mathbb{Z}}/2}
    \arrow["\operatorname{rank}", from=1-3, to=3-3]
    \arrow["\operatorname{rank}", from=1-1, to=3-1]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-1, to=1-3]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXEdXKGspIl0sWzMsMF0sWzIsMCwiVyhrKSJdLFswLDIsIlxcWloiXSxbMiwyLCJcXFpaLzIiXSxbMiw0LCJcXHJhbmsiXSxbMCwzLCJcXHJhbmsiXSxbMyw0XSxbMCwyXSxbMCw0LCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=}{Link
to Diagram}
\end{quote}

We can also write \({\operatorname{GW}}(k)\) in terms of generators
\(\left\langle{a}\right\rangle\) where
\(a\in k^{\times}/ (k^{\times})^{\times 2}\), where
\(\left\langle{a}\right\rangle\) is associated to a bilinear form
\begin{align*}
\left\langle{a}\right\rangle: k^{\times 2} &\to k \\
(x, y) &\mapsto axy
,\end{align*}
subject to relations

\begin{itemize}
\tightlist
\item
  \(\left\langle{a}\right\rangle\left\langle{b}\right\rangle = \left\langle{ab}\right\rangle\)
\item
  \(\left\langle{u}\right\rangle + \left\langle{v}\right\rangle = \left\langle{uv(u+v)}\right\rangle + \left\langle{u+v}\right\rangle\)
\item
  \(\left\langle{u}\right\rangle + \left\langle{-u}\right\rangle = \left\langle{1}\right\rangle + \left\langle{-1}\right\rangle = q_{\operatorname{hyp}}\),
  which is the matrix
  \begin{align*}
  q_{\operatorname{hyp}}= {
  \begin{bmatrix}
    {0} & {1} 
  \\
    {1} & {0}
  \end{bmatrix}
  }
  .\end{align*}
\end{itemize}

\end{remark}

\begin{example}[of known $\GW$ groups]

The \textbf{signature} is the difference between the numbers of positive
and negative ones in the associated matrix, and one can show
\begin{align*}
\operatorname{rank}: {\operatorname{GW}}({\mathbb{C}}) &{ { \, \xrightarrow{\sim}\, }}{\mathbb{Z}}\\ 
(\operatorname{rank}, \operatorname{sig}): {\operatorname{GW}}({\mathbb{R}}) &{ { \, \xrightarrow{\sim}\, }}{\mathbb{Z}}^{\times 2} \\
(\operatorname{rank}, {\operatorname{disc}}): {\operatorname{GW}}({\mathbb{F}}_q) &{ { \, \xrightarrow{\sim}\, }}{\mathbb{Z}}\times{\mathbb{F}}_q^{\times}/ ({\mathbb{F}}_q^{\times})^{\times 2}
\end{align*}
where the last is a situation where we can compute étale cohomology.

\end{example}

\begin{example}[Springer's theorem]

Let \(k\in \mathsf{Field}\) be complete and discretely valued with
residue field \(\kappa\), e.g.~\(k= { {\mathbb{Q}}_p }\) or
\({\mathbb{F}}_p{\left(\left( t \right)\right) }\) with
\(\kappa = {\mathbb{F}}_p\). Then if \(\operatorname{ch}k \neq 2\),
\begin{align*}
GW(k) { { \, \xrightarrow{\sim}\, }}W(k){ {}^{ \scriptscriptstyle\oplus^{2} }  }
.\end{align*}

\end{example}

\begin{remark}

For \(E_{/ {k}}\) a finite separable field extension, we'll have
transfers
\begin{align*}
\operatorname{Tr}_{E_{/ {k}} }: {\operatorname{GW}}(E) &\to {\operatorname{GW}}(k) \\
(V{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{\beta} E) &\mapsto (V{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{\beta} E \xrightarrow{\operatorname{Tr}_{E_{/k}}} k)
,\end{align*}
which coincide with classical transfers for field extensions.

\end{remark}

\begin{remark}

For Lannes/Morel's formula, given
\({\mathbb{P}}^1_{/ {k}} \xrightarrow{f} {\mathbb{P}}^1_{/ {k}}\) and
\(p\in {\mathbb{P}}^1_{/ {k}}\), we can write
\(f^{-1}(p) = \left\{{{ {q}_1, {q}_2, \cdots, {q}_{N}}}\right\}\) and
suppose \(J(q_i) = f'(q_i) \neq 0\) for all \(i\). Then we remember the
entire Jacobian and set
\begin{align*}
\deg f \coloneqq\sum_{i=1}^N \operatorname{Tr}_{k(q_i)_{/ {k}} } \left\langle{J(q_i)}\right\rangle
,\end{align*}
which in fact doesn't depend on \(p\). Morel defines an
\({\mathbb{A}}^1{\hbox{-}}\)degree
\begin{align*}
\deg^{{\mathbb{A}}^1}: [{\mathbb{P}}^n/{\mathbb{P}}^{n+1}, {\mathbb{P}}^n/{\mathbb{P}}^{n+1}]^{{\mathbb{A}}^1} \to {\operatorname{GW}}(k)
,\end{align*}
where we are taking unstable \({\mathbb{A}}^1{\hbox{-}}\)homotopy
classes of maps. Noting that an element of
\({\operatorname{GW}}({\mathbb{R}})\) was determined by its rank and
signature, we get a commutative diagram showing that
\(\deg^{{\mathbb{A}}^1}\) is compatible with rank, signature, and the
classical algebraic topological degree. There are other ways of
computing this degree besides taking the above sum: Cazanave,
\href{https://arxiv.org/abs/2103.16614}{Brazelton-McKean-Pauli} give
formulas in terms of Bézoutians.

\end{remark}

\hypertarget{homotopy}{%
\subsection{Homotopy}\label{homotopy}}

\begin{remark}

Recall that
\begin{align*}
X\wedge Y \coloneqq{ X\times Y \over (X\times{\operatorname{pt}}) \cup({\operatorname{pt}}\times Y)} && \in {\mathsf{Top}}_*
,\end{align*}
and \(S^n \wedge S^m \xrightarrow{\sim} S^{n+m}\) and
\((S^1){ {}^{ \scriptscriptstyle\wedge^{n} } } \xrightarrow{\sim} S^n\),
so we define
\begin{align*}
{\Sigma}_{S^1} X \coloneqq S^1 \wedge X
.\end{align*}
In \({\mathbb{A}}^1\) homotopy theory we declare
\({\mathbb{A}}^1 \simeq{\operatorname{pt}}\).

\end{remark}

\begin{example}[?]

We can take a pushout of the following form:

\begin{center}
\begin{tikzcd}
    & z && z \\
    z & {{\mathbb{G}}_m} && {{\mathbb{A}}^1\simeq{\operatorname{pt}}} \\
    \\
    {1/z} & {{\operatorname{pt}}\simeq{\mathbb{A}}^1} && { \therefore {\mathbb{P}}^1 \simeq{\Sigma}_{S^1} {\mathbb{G}}_m}
    \arrow[maps to, from=2-1, to=4-1]
    \arrow[from=2-2, to=4-2]
    \arrow[from=2-2, to=2-4]
    \arrow[maps to, from=1-2, to=1-4]
    \arrow[from=4-2, to=4-4]
    \arrow[from=2-4, to=4-4]
    \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-4, to=2-2]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMSwxLCJcXEdHX20iXSxbMSwzLCJcXHB0IFxcaG9tb3RvcGljIFxcQUFeMSJdLFswLDEsInoiXSxbMCwzLCIxL3oiXSxbMywxLCJcXEFBXjFcXGhvbW90b3BpYyBcXHB0Il0sWzEsMCwieiJdLFszLDAsInoiXSxbMywzLCJcXFBQXjEgXFxob21vdG9waWMgXFxTdXNwZW5kX3tTXjF9IFxcR0dfbSJdLFsyLDMsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbMCwxXSxbMCw0XSxbNSw2LCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzEsN10sWzQsN10sWzcsMCwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d}{Link
to Diagram}
\end{quote}

Here the formalism of homotopy pushouts allows us to conclude that in an
appropriate \({\mathbb{A}}^1{\hbox{-}}\)homotopy category,
\begin{align*}
{\Sigma}_{S^1} {\mathbb{G}}_m \coloneqq S^1\wedge{\mathbb{G}}_m \simeq{\mathbb{P}}^1
.\end{align*}

\end{example}

\begin{remark}[on motivic spheres]

We have
\begin{align*}
{\mathbb{G}}_m \coloneqq\operatorname{Spec}k[z, 1/z] = {\mathbb{A}}^1\setminus\left\{{{\operatorname{pt}}}\right\}
.\end{align*}
By taking pushouts inductively we can realize
\begin{align*}
{\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} \simeq{\Sigma}_{S^1} ({\mathbb{A}}^1\setminus\left\{{{\operatorname{pt}}}\right\}) \wedge({\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\}) \simeq(S^1){ {}^{ \scriptscriptstyle\wedge^{n-1} }  } \wedge({\mathbb{G}}_m){ {}^{ \scriptscriptstyle\wedge^{n} }  }
.\end{align*}

\end{remark}

\begin{remark}

We can use this to write
\begin{align*}
{\mathbb{P}}^n/{\mathbb{P}}^{n-1} 
&\simeq{{\mathbb{P}}^n \over {\mathbb{P}}^n\setminus\left\{{{\operatorname{pt}}}\right\} }\\
&\simeq{{\mathbb{A}}^n \over {\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} }\\
&\simeq{{\operatorname{pt}}\over {\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} }\\
&\simeq{\Sigma}_{S^1} \qty{ {\mathbb{A}}^n\setminus\left\{{{\operatorname{pt}}}\right\} } \\
&\simeq(S^1){ {}^{ \scriptscriptstyle\wedge^{n} }  } \wedge({\mathbb{G}}_m){ {}^{ \scriptscriptstyle\wedge^{n} }  }
.\end{align*}

\end{remark}

\begin{remark}

Stable homotopy shows that inverting \({\Sigma}\) is useful, which we
also do in the \({\mathbb{A}}^1{\hbox{-}}\)setting by inverting
\({\Sigma}_{{\mathbb{P}}^1} ({-}) \coloneqq{\mathbb{P}}^1 \wedge({-})\)
to obtain a stable homotopy category \({\mathsf{SH}}(k)\).

\end{remark}

\begin{theorem}[Morel, Hopkins-Morel]

For \(k\in \mathsf{Field}\), stably we have
\begin{align*}
[S^0, S^0] \xrightarrow{\sim} [{\mathbb{P}}^n/{\mathbb{P}}^{n-1}, {\mathbb{P}}^n/{\mathbb{P}}^{n-1}] \xrightarrow{\sim} {\operatorname{GW}}(k)
.\end{align*}
Moreover, there is a ring structure on homotopy classes which yields an
isomorphism of rings into \textbf{Milnor-Witt \(K{\hbox{-}}\)theory},
\begin{align*}
\bigoplus_{n\in {\mathbb{Z}}} [ S^0, {\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{n} }  }] \xrightarrow{\sim}  {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*(k)
.\end{align*}

\end{theorem}

\begin{remark}

\({\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*\) is a graded
associative algebra with generators
\([u] \in {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _1(k)\) for
\(u\in k^{\times}\) and
\(\eta \in {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _{-1}(k)\),
with relations

\begin{itemize}
\tightlist
\item
  \([u][1-u] = 0\), the \emph{Steinberg relations},
\item
  \([ab] = [a] + [b] + \eta[a][b]\),
\item
  \([a]\eta = \eta[a]\),
\item
  \(\eta q_{\operatorname{hyp}}= 0\) for
  \(q_{\operatorname{hyp}}\coloneqq\eta[-1] + 2\)
\end{itemize}

\end{remark}

\begin{remark}

There is an isomorphism
\begin{align*}
{\operatorname{GW}}(k) &\xrightarrow{\sim}  {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _0(k) \\
\left\langle{a}\right\rangle &\rightleftharpoons 1 + \eta[a] \\
q_{\operatorname{hyp}}\coloneqq\left\langle{1}\right\rangle + \left\langle{-1}\right\rangle &\rightleftharpoons 1 + 1 + \eta[-1]
.\end{align*}

\end{remark}

\begin{remark}[on the proof]

\([a]\) yields a map
\begin{align*}
[a]: S^0 = (\operatorname{Spec}k)^{\coprod 2} &\to {\mathbb{G}}_m \\
p &\mapsto a
,\end{align*}
where \(p\) is the non-basepoint, and
\begin{align*}
\eta: {\mathbb{A}}^2\setminus\left\{{{\operatorname{pt}}}\right\} &\to {\mathbb{P}}^1 \\
(x, y) &\mapsto [x: y]
.\end{align*}
On \({\mathbb{C}}{\hbox{-}}\)points,
\({\mathbb{C}}^2\setminus\left\{{0}\right\}\simeq S^3\) maps to
\({\mathbb{CP}}^1\simeq S^2\) by the Hopf map, but on
\({\mathbb{R}}{\hbox{-}}\)points we get
\(S^1 \xrightarrow{\deg = -2} S^1\) implying that \(\eta\) is not
nilpotent, which is a new fact.

\end{remark}

\begin{remark}

We can define
\begin{align*}
X\vee Y = {X \times Y \over {\operatorname{pt}}_X \sim {\operatorname{pt}}_Y}
\end{align*}
and get maps
\begin{align*}
X\vee Y \to X\times Y\to X\wedge Y
.\end{align*}
which yields
\begin{align*}
{\Sigma}(X\times Y) \xrightarrow{\sim } {\Sigma}X \vee{\Sigma}Y \vee{\Sigma}(X\wedge Y)
.\end{align*}

\end{remark}

\begin{lemma}[?]

In \({\mathsf{SH}}(k)\),we get

\begin{center}
\begin{tikzcd}
    {{\mathbb{G}}_m{ {}^{ \scriptscriptstyle\times^{2} }  } } && {{\mathbb{G}}_m} \\
    \\
    { {{\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{2} }  }} \vee{{\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{2} }  }} } && {{\mathbb{G}}_m}
    \arrow["{\operatorname{mult}}", from=1-1, to=1-3]
    \arrow["{(1, 1, \eta)}", from=3-1, to=3-3]
    \arrow["\simeq", from=1-1, to=3-1]
    \arrow[equal, from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXEdHX21ee1xcdGltZXMgMn0iXSxbMCwyLCJcXEdHX21ee1xcd2VkZ2Vwcm9kIDJ9IFxcd2VkZ2Vwcm9kIFxcR0dfbV57XFxzbWFzaHByb2QgMn0iXSxbMiwyLCJcXEdHX20iXSxbMiwwLCJcXEdHX20iXSxbMCwzLCJcXG11bHQiXSxbMSwyLCIoMSwgMSwgXFxldGEpIl0sWzAsMSwiXFxzaW1lcSIsMV0sWzMsMiwiXFxzaW1lcSIsMV1d}{Link
to Diagram}
\end{quote}

\end{lemma}

\begin{lemma}[?]

The map
\begin{align*}
f: {\mathbb{P}}^1 &\to {\mathbb{P}}^1 \\
z &\mapsto az
\end{align*}
is equal to \(1 + \eta[a]\) in \({\mathsf{SH}}(k)\), since
\(f = {\Sigma}g\) where
\begin{align*}
g: {\mathbb{G}}_m &\to {\mathbb{G}}_m \\ 
z &\mapsto az
,\end{align*}
which is equal to
\begin{align*}
{\Sigma}({\mathbb{G}}_m \times k \xrightarrow{1\times a} {\mathbb{G}}_m^{\times 2} \xrightarrow{{\operatorname{mult}}} {\mathbb{G}}_m)
.\end{align*}

\end{lemma}

\begin{remark}

The lemma implies the relation \([ab] = [a] + [b] + \eta[a][b]\), and it
turns out there's an isomorphism to motivic homotopy groups of spheres:
\begin{align*}
 {\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*(k) \xrightarrow{\sim} \bigoplus _{n\in {\mathbb{Z}}} [S^0, {\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{n} }  }]
.\end{align*}

\end{remark}

\hypertarget{big-problems}{%
\subsection{Big Problems}\label{big-problems}}

\begin{remark}

Notation: we'll write
\begin{align*} 
\bigoplus_{n\in {\mathbb{Z}}} \pi_{n, n} {\mathbb{S}}\coloneqq\bigoplus _{n\in {\mathbb{Z}}} [S^0, {\mathbb{G}}_m{ {}^{ \scriptscriptstyle\wedge^{n} }  } ]
\end{align*}
to be the zero line of the homotopy groups of spheres, and generally
\(\bigoplus _n \pi_{n+r, n} {\mathbb{S}}\) for the \(r{\hbox{-}}\)line.
Classical homotopy groups of spheres encode interesting geometric
information, and we're finding that the corresponding motivic homotopy
groups of spheres do as well.
\href{https://arxiv.org/abs/1604.00365}{Röndigs-Spitzweck-Østvær}
compute the 1-line for \(\operatorname{ch}k\neq 2\) in a 2019 Annals
paper, and we have some information about the 2-line.

\end{remark}

\begin{question}

What is
\([{\mathbb{P}}^n/{\mathbb{P}}^{n-1}, {\mathbb{P}}^n/{\mathbb{P}}^{n-1}]\)
for more general rings?
\href{https://arxiv.org/abs/2102.01618}{Bachmann-Østvær (2021)} do this
over \({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{2} } \right] }\) and
show
\begin{align*}
\pi_{0,0}{\mathbb{S}}\otimes{ {\mathbb{Z}}_{\widehat{2}} } \xrightarrow{\sim} {\operatorname{GW}}({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{2} } \right] })\otimes{ {\mathbb{Z}}_{\widehat{2}} }
.\end{align*}

\end{question}

\begin{question}

What is \(\pi_{*, *}{\mathbb{S}}\) in general?

\end{question}

\begin{question}

Is there a Freudenthal suspension theorem? I.e. which stable elements of
\(\pi_{*, *} {\mathbb{S}}\) correspond to unstable groups?

\end{question}

\hypertarget{counting-things}{%
\subsection{Counting Things}\label{counting-things}}

\begin{remark}

Many people have used the \({\mathbb{A}}^1{\hbox{-}}\)Euler class for
interesting things! Let \(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\)
with \(\dim X = d\) and let
\(V\to X \in { { {\mathsf{Bun}}_{\operatorname{GL}_r} }}_{/ {X}}\) be a
vector bundle.

\end{remark}

\begin{definition}[Orientation of bundles]

A bundle \(V\to X\) is \textbf{oriented} by the following data:
\((L, \phi)\) where \(L\to X\) is a line bundle and \(\phi\) is a
trivialization
\begin{align*}
\phi: \operatorname{det}V { { \, \xrightarrow{\sim}\, }}L^{\otimes 2}
.\end{align*}
It is \textbf{relatively oriented} when
\(\mathop{\mathrm{Hom}}(\operatorname{det}TX, \operatorname{det}V)\) is
oriented, where
\(\operatorname{det}({-}) = \bigwedge\nolimits^{\text{top}}({-})\).

\end{definition}

\begin{example}[?]

For \(X = {\mathbb{P}}^n\) or \({\operatorname{Gr}}(m, n)\)
(parameterizing copies of \({\mathbb{P}}^m\) in \({\mathbb{P}}^n\)),
then
\begin{align*}
\omega_x = \operatorname{det}T^n X = {\mathcal{O}}(-n-1)
,\end{align*}
and \(X\) is orientable iff \(n\) is odd. For \({\mathbb{P}}^1\),
\({\mathcal{O}}(n)\) is relatively orientable iff \(n\) is even.

\end{example}

\hypertarget{euler-numbers}{%
\subsection{Euler Numbers}\label{euler-numbers}}

\begin{definition}[Euler Number in $\GW(k)$]

Suppose \(X\in { {\mathsf{sm}}{\mathsf{Sch}}}_{/ {k}}\) is proper with
\(\dim X = d\) and consider a vector bundle

\begin{center}
\begin{tikzcd}
\operatorname{GL}_d \curvearrowright F
  \ar[r] 
& 
V
  \ar[d] 
\\
& 
X 
\ar[u, "f", bend left, dotted]
\end{tikzcd}
\end{center}

Suppose

\begin{itemize}
\tightlist
\item
  \(V\) is relatively oriented by \((L, \phi)\), and \(f\) is a section
  with isolated zeros, so \(\left\{{f=0}\right\}\) consists of zeros of
  multiplicity 1, or equivalently
\item
  For all \(x\in \left\{{f = 0}\right\}\), the composite map
  \begin{align*}
  Tf \coloneqq\qty{T_x X \to T_{f(x)} V \xrightarrow{\sim} T_x X \oplus V_x \xrightarrow{p_2} V_x} && f(x) \coloneqq(x, 0)
  \end{align*}
  has nonvanishing determinant.
\end{itemize}

Then the \textbf{Euler number of \((V, \phi)\)} with respect to \(f\) is
defined as
\begin{align*}
n(V, \phi, f) \coloneqq\sum_{x \in \left\{{f = 0 }\right\} \subseteq X} \deg_x f
.\end{align*}
where \(\deg_x f\) can be computed by

\begin{itemize}
\tightlist
\item
  Choosing local Nisnevich coordinates on \(X\) ,
\item
  Choosing local trivializations of \(V\) which are ``compatible'' with
  \(\phi\),
\end{itemize}

Then locally writing
\begin{align*}
f: {\mathbb{A}}^d \to {\mathbb{A}}^d \implies J_f \coloneqq\operatorname{det}\qty{{\partial}f_i \over {\partial}x_j}
,\end{align*}
one has
\begin{align*}
\text{For } J_f(x) \neq 0 \in \kappa(x), \quad
\deg_x f \coloneqq\operatorname{Tr}_{\kappa(x) _{/ {k}} } \left\langle{J_f(x)}\right\rangle
.\end{align*}

\end{definition}

\begin{remark}

Equivalently, \(T_x f\in \mathop{\mathrm{Hom}}(T_x X, V_x)\) and we can
define
\begin{align*}
J_f(x) \coloneqq\operatorname{det}T_x f \in \mathop{\mathrm{Hom}}(\operatorname{det}T_x X, \operatorname{det}V_x) { { \, \xrightarrow{\sim}\, }}L_x{ {}^{ \scriptstyle\otimes_{k}^{2} }  }
,\end{align*}
where the orientation provides the isomorphism. Picking any basis for
\(L_x{ {}^{ \scriptstyle\otimes_{k}^{2} } }\) yields a number which is
well-defined in \(\kappa(x) / (\kappa(x)^{\times})^2\) by choosing a
trivialization of \(L_x\).

\end{remark}

\begin{question}

What happens if the zeros of \(f\) have multiplicity \(m_i > 1\)? In the
classical setting, we didn't say what happens when \(J_f(x) = 0\). We'll
answer this next time.

\end{question}

\begin{question}

Why is the Euler number \(n(V, f)\) independent of the section \(f\)?
Analogously, why is the number of intersections in the original problem
2, not depending on which specific lines were chosen?

\end{question}

\begin{answer}

Sections with isolated zeros are often connected by 1-parameter
\({\mathbb{A}}^1{\hbox{-}}\)families of such sections, and
\({\operatorname{GW}}(k[x]) \xrightarrow{\sim} {\operatorname{GW}}(k)\),
although this is hard to show.

Alternatively, the Euler number is a pushforward of an Euler class
taking values in interesting cohomology theories, so
\(n(V, f) = \pi_* e(V, f)\).

\end{answer}

\hypertarget{kirsten-wickelgren-talk-2-friday-july-16}{%
\section{Kirsten Wickelgren, Talk 2 (Friday, July
16)}\label{kirsten-wickelgren-talk-2-friday-july-16}}

\hypertarget{intro-6}{%
\subsection{Intro}\label{intro-6}}

\begin{remark}

Recall that we have a classical degree map
\begin{align*}
\deg: [S^n, S^n] \to {\mathbb{Z}}
\end{align*}
which roughly counts preimages. Given
\(f\in \mathop{\mathrm{Hom}}_{\mathsf{Top}}(S^n, S^n)\) and
\(p\in S^n\), we write
\(f^{-1}(p) = \left\{{ { {q}_1, {q}_2, \cdots, {q}_{N}} }\right\}\) and
have a formula \(\deg f = \sum \deg_{q_i} f\) where the local degrees
\(\deg_{q_i}f\) can be computed by picking orientation-compatible
coordinates \({ \left\{{ {x}_1, {x}_2, \cdots, {x}_{n} }\right\} }\)
near \(q_i\) and
\({ \left\{{ {y}_1, {y}_2, \cdots, {y}_{n} }\right\} }\) near \(p\). In
these coordinates we can form the Jacobian
\(J_f \coloneqq\operatorname{det}{\frac{\partial f_i}{\partial x_j}\,}\)
and write
\begin{align*}
\deg_x f =
\begin{cases}
+1 & J(q_i) > 0 
\\
-1 & J(q_i) < 0.
\end{cases}
\end{align*}

\end{remark}

\begin{question}

What happens if the zeros of \(f\) are not of multiplicity 1, so
\(J_f(q_i) = 0\)?

\end{question}

\begin{proposition}[Eisenbud–H.Levine–Khimshiashvili signature formula]

Over \(k\coloneqq{\mathbb{R}}\), any quadratic form can be diagonalized
to \(\operatorname{diag}(1,\cdots,1, -1,\cdots,-1)\), and there is a
formula
\begin{align*}
\deg_x f = \operatorname{sig}\omega^{{\mathrm{EKL}}}
\end{align*}
where \(\omega^{{\mathrm{EKL}}}\) is the isomorphism class of the
bilinear form defined in the following way: for
\(f = (f_1, \cdots, f_n)\), set
\begin{align*}
Q \coloneqq{{\mathbb{R}}[x_1, \cdots, x_n]_0 \over \left\langle{{ {f}_1, {f}_2, \cdots, {f}_{n}} }\right\rangle }
\end{align*}
which is a finite dimensional local complete intersection. Since \(Q\)
is Gorenstein\footnote{The dualizing sheaf is locally free}, there is an
isomorphism \(\mathop{\mathrm{Hom}}_k(Q, k) \xrightarrow{\sim} Q\),
which we can take to be the bilinear form.\footnote{Even better, there
  is a distinguished isomorphism coming from a distinguished socle
  element (Scheja-Storch).}

\end{proposition}

\begin{remark}

This form can be made very explicit: writing
\(J_f = \operatorname{det}{\frac{\partial f_i}{\partial x_j}\,} \in Q\),
choose a \(k{\hbox{-}}\)linear map \(\eta: Q\to k\) such that
\(\eta(J_f) = \dim_k Q\) and set
\begin{align*}
\omega^{{\mathrm{EKL}}} &\coloneqq\qty{ Q{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{{\operatorname{mult}}} Q \xrightarrow{\eta} k} \\ \\
\implies \omega^{{\mathrm{EKL}}}: Q{ {}^{ \scriptstyle\otimes_{}^{2} }  } &\to k \\
(g, h) &\mapsto \eta(gh)
.\end{align*}
It turns out that the isomorphism class of \(\omega^{{\mathrm{EKL}}}\)
does not depend on the choice of \(\eta\).

\end{remark}

\begin{example}[?]

Let \(f:{\mathbb{A}}^1\to {\mathbb{A}}^1\) and \(f(z) = z^2\) with
\(q=0\). Then
\begin{align*}
Q = {k[x]_0 \over \left\langle{x^1}\right\rangle } \xrightarrow{\sim} {k[x]\over \left\langle{x^2}\right\rangle}
\end{align*}
and \(J_f = 2x\). We then get
\(\omega^{{\mathrm{EKL}}} = { \begin{bmatrix}  {0} & {1} \\  {1} & {0} \end{bmatrix} }\),
which up to a change of basis is
\(h \coloneqq{ \begin{bmatrix}  {1} & {0} \\  {0} & {-1} \end{bmatrix} }\).

\end{example}

\begin{question}

Eisenbud notes that \(\omega^{\mathrm{EKL}}\) is defined over fields of
arbitrary characteristic not equal to 2, does it have a topological
interpretation?

\end{question}

\begin{remark}

Yes! It comes from the \({\mathbb{A}}^1{\hbox{-}}\)degree.

\end{remark}

\begin{theorem}[Kass-W.]

\(\omega^{{\mathrm{EKL}}} = \deg_q^{{\mathbb{A}}^1}f\) is the local
degree in \({\operatorname{GW}}(k)\) when \(\kappa(q) = k\).
\href{https://arxiv.org/pdf/1912.04788.pdf}{Brazelton, Burklund, Mckean,
Montoro, Opie} handle the case when \(\kappa(q)/k\) is separable.

\end{theorem}

\hypertarget{mathbba1hbox-milnor-numbers}{%
\subsection{\texorpdfstring{\({\mathbb{A}}^1{\hbox{-}}\)Milnor
numbers}{\{\textbackslash mathbb\{A\}\}\^{}1\{\textbackslash hbox\{-\}\}Milnor numbers}}\label{mathbba1hbox-milnor-numbers}}

\begin{definition}[Node, hypersurface singularity]

For \(\operatorname{ch}k\neq 2\), the simplest type of singularity is a
\textbf{node}, defined over
\(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) as a
point \(p\in X\) such that\footnote{Her \({\mathcal{O}}_{X, p}\) is the
  stalk of the structure sheaf at \(p\), which is a local ring with a
  unique maximal ideal \({\mathfrak{m}}_p\), and the LHS is completion
  at that ideal, so
  \begin{align*}
  \widehat{{\mathcal{O}}_{X, p}} \coloneqq\qty{{\mathcal{O}}_{X, p}}{ {}_{ \widehat{{\mathfrak{m}}_p} } }
  .\end{align*}
}
\begin{align*}
\widehat{{\mathcal{O}}_{X, p}}
\xrightarrow{\sim} 
{ \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu  { \left[ { { {x}_1, {x}_2, \cdots, {x}_{n}} } \right] }  \over \sum x_i^2 + \text{h.o.t.} }
,\end{align*}

A \textbf{hypersurface singularity} is a point
\(p\in \left\{{f=0}\right\} \subseteq X\).

\end{definition}

\begin{definition}[Milnor number]

Let \(k\coloneqq{\mathbb{C}}\). If you vary \(X\) in a family
\begin{align*}
X_+ \coloneqq\left\{{f(x_1,\cdots, x_n) + \sum a_i x_i = t}\right\}
,\end{align*}
then the singularity \(p\) bifurcates into nodes. The number of nodes is
given by the \textbf{Milnor number}, defined as \(M_p\), the number of
nodes in the family \(X_+\) for any sufficiently small
\(\left\{{a_i}\right\}\). For \(R={\mathbb{C}}\), this is explicitly
described as
\begin{align*}
M_p \coloneqq\deg_p^{{\mathsf{Top}}}\operatorname{grad}f
.\end{align*}

\end{definition}

\begin{remark}

For other \(k\) with \(\operatorname{ch}k \neq 2\), nodes come in
different types: given a residue field \(L\) at a node \(p\), the
tangent directions defined over some extension \(L[\sqrt a]\) for
\(a\in L^{\times}/ (L^{\times}){ {}^{ \scriptscriptstyle\times^{2} } }\).

\end{remark}

\begin{example}[?]

Over \(k={\mathbb{R}}\), one has examples like

\begin{itemize}
\item
  \(x_1^2 +x_2^2 = 0\), yielding a non-split node and non-rational
  tangent directions

  \includegraphics{figures/2021-07-19_19-58-25.png}
\item
  \(x_1^2-x_2^2=0\), yielding a split node and rational tangent
  directions

  \includegraphics{figures/2021-07-16_13-26-04.png}
\end{itemize}

\end{example}

\begin{definition}[Type of a node]

Let \(p\) be a node with
\begin{align*}
\widehat{{\mathcal{O}}_{X, p}} { { \, \xrightarrow{\sim}\, }}{ L { \left[\left[ { { {x}_1, {x}_2, \cdots, {x}_{n}} } \right] \right] }  \over \sum a_i x_i^2 }, 
&& L \coloneqq\kappa(p)
.\end{align*}
The \textbf{type} \(p\) is defined as
\begin{align*}
{\operatorname{type}}(p) \coloneqq\operatorname{Tr}_{L_{/ {k}} } \left\langle{2^n \prod_{i=1}^n a_i}\right\rangle \in {\operatorname{GW}}(k)
.\end{align*}

\end{definition}

\begin{definition}[$\AA^1\dash$Milnor numbers]

The \textbf{\({\mathbb{A}}^1{\hbox{-}}\)Milnor number} is defined as
\begin{align*}
M_p \coloneqq\deg_p^{{\mathbb{A}}_1} \operatorname{grad}f = \sum_{p\in N} {\operatorname{type}}(p)
,\end{align*}
\(N\) is the set of nodes of \(f\) in a family for a generic
\(\left\{{{ {a}_1, {a}_2, \cdots, {a}_{n}} }\right\}\).

\begin{quote}
Note: the second equality is due to Kass-Wickelgren.
\end{quote}

\end{definition}

\begin{example}[?]

Let \(f(x,y)\coloneqq x^3-y^2\) with \(\operatorname{ch}k\neq 2,3\),
then

\begin{itemize}
\tightlist
\item
  \(p=(0, 0) \in \left\{{f=0}\right\}\),
\item
  \(\nabla f = (3x^2, -2y)\), and
  \begin{align*}
  \deg^{{\mathbb{A}}^1} \nabla f 
  &= \deg^{{\mathbb{A}}^1} (x\mapsto 3x^2 ) \deg^{{\mathbb{A}}^1} (y\mapsto -2y) \\
  &= {
  \begin{bmatrix}
    {0} & {1/3} 
  \\
    {1/3} & {0}
  \end{bmatrix}
  } \left\langle{-2}\right\rangle \\
  &= \left\langle{1}\right\rangle + \left\langle{-1}\right\rangle \\
  &= q_{{\operatorname{hyp}}}
  ,\end{align*}
  which has rank 2 and thus \(M = 2\). This yields a cusp.
\end{itemize}

The family \(y^2 = x^3 + ax + t\) for \(a\neq 0\) yields a family:

\includegraphics{figures/2021-07-16_13-33-13.png}

\begin{itemize}
\tightlist
\item
  In the first term, the cusp bifurcates into 2 nodes, yields
  \(\operatorname{rank}M^{{\mathbb{A}}^1}(C) = M(C)\) for \(C\) the cusp
  curve
\item
  In the second, there are singular fibers when \(x^3+ax+t\) has double
  roots, which happens when the discriminant is zero, so this occurs iff
  \(-4a^3 -27b^2 =0\).
\end{itemize}

This yields a bifurcation into two nodes, and for example,

\begin{itemize}
\tightlist
\item
  Over \({\mathbb{F}}_5\):
  \(\left\langle{1}\right\rangle = \left\langle{-1}\right\rangle\), so
  this can not bifurcate into 1 split and 1 non-split rational node.
\item
  Over \({\mathbb{F}}_7\):
  \(\left\langle{1}\right\rangle \neq \left\langle{-1}\right\rangle\),
  so this can not bifurcate into 2 split or 2 non-split nodes.
\end{itemize}

\end{example}

\begin{remark}

The classical Milnor number appears in conductor formulas and are
related to the Euler characteristic \(\chi\) of the \emph{Milnor fiber}.
M. Levine-Lehalleur-Srinivas and R. Azouri have subtle quadratic
enrichments to \({\operatorname{GW}}(k)\).

\end{remark}

\hypertarget{mathbba1hbox-euler-characteristic-chimathbba1}{%
\subsection{\texorpdfstring{\({\mathbb{A}}^1{\hbox{-}}\)Euler
characteristic
\(\chi^{{\mathbb{A}}^1}\)}{\{\textbackslash mathbb\{A\}\}\^{}1\{\textbackslash hbox\{-\}\}Euler characteristic \textbackslash chi\^{}\{\{\textbackslash mathbb\{A\}\}\^{}1\}}}\label{mathbba1hbox-euler-characteristic-chimathbba1}}

\begin{definition}[Relatively oriented bundles]

For
\(X\in {\mathsf{sm}}{\mathsf{Var}}^{\mathop{\mathrm{proj}}}_{/ {k}}\),
we saw last time that a vector bundle \(V\to X\) is \textbf{relatively
oriented} by \((L, \phi)\) where \(L\to X\) is a line bundle and
\begin{align*}
\phi: L{ {}^{ \scriptstyle\otimes_{k}^{2} }  } \xrightarrow{\sim} \mathop{\mathrm{Hom}}(\operatorname{det}{\mathsf{T}}X, \operatorname{det}V)
.\end{align*}

\end{definition}

\begin{remark}

The tangent bundle \({\mathsf{T}}X\) has a canonical relative
orientation since
\begin{align*}
\mathop{\mathrm{Hom}}(\operatorname{det}{\mathsf{T}}X, \operatorname{det}{\mathsf{T}}X) \xrightarrow{\sim} {\mathcal{O}}\xrightarrow{\sim} {\mathcal{O}}{ {}^{ \scriptstyle\otimes_{}^{2} }  }
\end{align*}
where \({\mathcal{O}}\) is a trivial bundle of rank 1. It follows that
we may define the Euler number of tangent bundle:
\begin{align*}
\chi^{{\mathbb{A}}^1}(X) \coloneqq n({\mathsf{T}}X) \in {\operatorname{GW}}(k)
,\end{align*}
where \(n\) is the Euler number.

\end{remark}

\begin{theorem}[M. Levine]

\(\chi^{{\mathbb{A}}^1}(X)\) is equal to the \textbf{categorical Euler
characteristic}. The definition is omitted here, see
\href{https://arxiv.org/abs/1703.03049}{Levine's ``Enumerative geometry
with quadratic forms''}.

\end{theorem}

\begin{example}[M. Levine-Lehalleur-Srinivas]

\envlist

\begin{quote}
See \url{https://arxiv.org/abs/2101.00482}.
\end{quote}

Let \(X \subseteq {\mathbb{P}}^{n+1}\) for \(n\) even, written
\(X \coloneqq\left\{{F=0}\right\}\) for
\(f\in {\mathbb{R}}[x_0, \cdots, x_n]_{e}\) homogeneous of degree \(e\).
Take
\begin{align*}
\omega^{{\mathrm{EKL}}}: Q{ {}^{ \scriptstyle\otimes_{}^{2} }  }\to k && 
Q \coloneqq{k[x_0,\cdots, x_n] \over \left\langle{ {\frac{\partial f}{\partial x_0}\,}, \cdots, {\frac{\partial f}{\partial x_n}\,} }\right\rangle }
.\end{align*}

Define \(B_{\operatorname{Jac}}\) to be the restriction of
\(\omega^{\mathrm{EKL}}\) to \(\bigoplus_{q=0}^n Q_{(q+1)e-n-2}\), then
\begin{align*}
\chi^{{\mathbb{A}}^1}(X) = \left\langle{e}\right\rangle + \left\langle{-e}\right\rangle B_{\operatorname{Jac}} + {n\over 2}q_{{\operatorname{hyp}}}
.\end{align*}

\end{example}

\begin{example}[Clebsch surface]

Take the Clebsch surface
\begin{align*}
C \coloneqq\left\{{ {\left[ {{ {x}_1, {x}_2, \cdots, {x}_{3}}} \right]} {~\mathrel{\Big|}~}\sum x_i^3 = \qty{\sum x_i}^3  }\right\} \subseteq {\mathbb{P}}^3_{/ {{\mathbb{Q}}}} 
.\end{align*}
Then \(\chi(C) = 9\), and
\begin{align*}
\chi^{{\mathbb{A}}^1}(C) = 2q_{{\operatorname{hyp}}} + \left\langle{-10}\right\rangle + \left\langle{-6}\right\rangle + \left\langle{-21}\right\rangle + \left\langle{-14}\right\rangle + \left\langle{-2}\right\rangle
,\end{align*}
which can be computed with Macaulay2!

\end{example}

\hypertarget{cohomology-and-the-euler-class}{%
\subsection{Cohomology and the Euler
Class}\label{cohomology-and-the-euler-class}}

\begin{remark}

Note that we have a stable \({\mathbb{A}}^1{\hbox{-}}\)homotopy category
\({\mathsf{SH}}(k)\), so we can take cohomology theories on
\(X\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {k}}\):

\begin{itemize}
\tightlist
\item
  \(H_{ \mathrm{mot}}\) or \(H{\mathbb{Z}}\): motivic cohomology
\item
  \(\tilde H_{ \mathrm{mot}}\) or \(H\tilde {\mathbb{Z}}\): extended
  motivic cohomology
\item
  \({\mathsf{K}}\): \({\mathsf{K}}{\hbox{-}}\)theory
\item
  \({\operatorname{KO}}\): Hermitian \({\mathsf{K}}{\hbox{-}}\)theory
\end{itemize}

It turns out that
\begin{align*}
H^n(X) = \pi_{-n}\mathop{\mathrm{Hom}}_{{\mathsf{SH}}(k)}(X, H) = [X, {\Sigma}^n H]
,\end{align*}
and it's useful to allow twisting the second term by shifts.

\end{remark}

\begin{definition}[Twisted suspension]

Letting \(V\to X\) be a vector bundle, we can define the \textbf{twisted
suspension}
\begin{align*}
H^V(X) \coloneqq[X, \mathop{\mathrm{Th}}(V) \wedge H]
&& \text{where } 
\mathop{\mathrm{Th}}(V) \coloneqq{V {}^{ \vee }\over V {}^{ \vee }\setminus\left\{{0}\right\}} = {{\mathbb{P}}(V {}^{ \vee }\oplus {\mathcal{O}}) \over {\mathbb{P}}(V {}^{ \vee })}
\end{align*}
where taking the dual is a condition needed for \(K{\hbox{-}}\)theory.
This reduces to a usual suspension for the trivial bundle.

\end{definition}

\begin{example}[?]

For
\(V\coloneqq({\mathcal{O}}_X){ {}^{ \scriptscriptstyle\oplus^{n} } }\)
the trivial rank \(n\) bundle on \(X\), we have \(H^V = H^n\).

\end{example}

\begin{example}[?]

\begin{align*}
H{\mathbb{Z}}^n(X) \cong H_{ \mathrm{mot}} ^{2n}(X; {\mathbb{Z}}(n)) \cong H^{2n, n}_{ \mathrm{mot}} (X) \cong {\operatorname{CH}}^n(X)
,\end{align*}
the Chow group of codimension \(n\) cycles modulo rational equivalence.
These are the \emph{geometric gradings}.

\end{example}

\begin{example}[?]

\(\tilde{H{\mathbb{Z}}}^n(X) \cong \tilde{{\operatorname{CH}}}^n(X)\)
the \textbf{Chow-Witt group}, also called the \textbf{oriented Chow
group}. Using the Gersten resolution, these can be expressed as formal
sums of codimension \(n\) subvarieties with coefficients in
\({\operatorname{GW}}(k(Z))\), where \(k(Z)\) are the rational functions
of \(Z\), subject to some conditions, modulo equivalence.

\end{example}

\begin{example}[?]

\({\mathsf{K}}^0(X)\) is the group completion of vector bundles on
\(X\), and \({\operatorname{KO}}^0(X)\) additionally remembers the data
of a nondegenerate symmetric bilinear form. These are ``representable''
in a sense because they are related to Thom spectra.

\end{example}

\begin{definition}[Cohomology with support]

Let \(Z\hookrightarrow X\) be a closed subscheme, then define the
\textbf{cohomology with support}
\begin{align*}
H_Z^V(X) \coloneqq { \left[ { { X \over X\setminus Z}, \mathop{\mathrm{Th}}(V) \wedge H} \right] } _{{\mathsf{SH}}(k)}
.\end{align*}

\end{definition}

\begin{definition}[Euler class]

For \(H\) a cohomology theory and \({\mathbb{S}}|to H\) a ring? and a
vector bundle \(V\to X\) with a section \(f\) (for example, the zero
section), then the \textbf{Euler class}
\begin{align*}
e^H(V, f) \in H_{\left\{{f=0}\right\}}^{V {}^{ \vee }}(X)
\end{align*}
is the class of the map
\begin{align*}
{X\over X\setminus\left\{{f=0}\right\}} \xrightarrow{f} {V\over V\setminus\left\{{0}\right\}} \wedge H
.\end{align*}

\end{definition}

\begin{definition}[Euler Number]

For \(f:X\to S\) is a \textbf{local closed immersion} (\(\mathrm{lci}\))
if it locally factors as \(U \xrightarrow{i} P \xrightarrow{p} S\) with
\(p\) smooth and \(i\) a closed immersion determined by a \emph{Koszul
regular sequence} (so modding out doesn't yield a zero divisor, and the
higher cohomology of the Koszul complex is zero).

\end{definition}

\begin{remark}

Some properties:

\begin{itemize}
\item
  This has a well-behaved \textbf{cotangent complex} \(L_f\).
\item
  For a regular embedding, \(L_i \simeq N_u {}^{ \vee }P[1]\) is
  \textbf{conormal bundle}.\footnote{I.e. the dual of the normal bundle.}
\item
  \(L_p \simeq\Omega_{P/S} \simeq{\mathsf{T}}_p {}^{ \vee }\).
\item
  \(L_{p_i}\) is determined by \(i^* L_p\to L_{p_i} \to L_i\),
\item
  There is a coherent Serre duality related to \(L_f\).
\end{itemize}

There is also a good notion of pushforward: let \(p:X\to S\) be proper
\footnote{Here \textbf{proper} means that the preimage of a compact set
  is compact.} and \(\mathrm{lci}\).

Then there is a \textbf{Becker-Gottlieb transfer}
\begin{align*}
\operatorname{Tr}^{\text{BG}}: {\Sigma}_+^\infty S\to \mathop{\mathrm{Th}}(L_p)
,\end{align*}
and the cartoon is the following:

\includegraphics{figures/2021-07-16_14-06-20.png}

Here we embed \(X\to S\) into the trivial bundle over \(S\) and take a
neighborhood. Letting \(B\) be the trivial bundle over \(S\), then
\(\mathop{\mathrm{Th}}(B) \cong \Sigma_+^\infty S\) and collapsing
fiberwise quotients by the complement of the neighborhood:

\includegraphics{figures/2021-07-16_14-08-44.png}

This yields \(p_*: H^{L_p}(X) \to H^0(S)\).

\end{remark}

\hypertarget{oriented-cohomology-theories-and-euler-numbers}{%
\subsection{Oriented Cohomology Theories and Euler
Numbers}\label{oriented-cohomology-theories-and-euler-numbers}}

\begin{remark}

\envlist

\begin{itemize}
\item
  \(H\) is \(\operatorname{GL}{\hbox{-}}\)oriented if
  \(H_Z^n(X) \xrightarrow{\sim} H_Z^V(X)\) with
  \(n\coloneqq\operatorname{rank}V\).

  An example is \(H{\mathbb{Z}}, {\mathsf{K}}\), but non examples are
  \({H\tilde {\mathbb{Z}}}, {\operatorname{KO}}\).
\item
  \(H\) is \({\operatorname{SL}}{\hbox{-}}\)oriented if

  \begin{itemize}
  \tightlist
  \item
    \(H_Z^V(X) \xrightarrow{\sim} H_Z^{V'}(X)\),
  \item
    \(\operatorname{rank}V = \operatorname{rank}V'\) and
  \item
    \(\operatorname{det}V \xrightarrow{\sim} \operatorname{det}V' \otimes L{ {}^{ \scriptstyle\otimes_{}^{2} } }\)
    for \(L\to X\) a line bundle.
  \end{itemize}

  An example is \(H\tilde{{\mathbb{Z}}}, {\operatorname{KO}}\).
\end{itemize}

\end{remark}

\begin{remark}

For \(V\to X\) a relatively oriented vector bundle on
\(X \xrightarrow{p} k\) with \(p\) smooth and proper, and \(H\) an
\({\operatorname{SL}}_c{\hbox{-}}\)oriented cohomology theory. Then
\begin{align*}
H^{V {}^{ \vee }}(X) \xrightarrow{\sim} H^{{\mathsf{T}} {}^{ \vee }X}(X)
.\end{align*}
Letting \(f\) be any section of \(V\), e.g.~the zero section, then

\begin{center}
\begin{tikzcd}
    {e^H(V, f) \in} & {H_{\left\{{f=0}\right\}}^{V {}^{ \vee }}(X)} &&& {H^{V {}^{ \vee }}(X)} \\
    & {X\to {X\over X\setminus\left\{{f=0}\right\}}} &&& {\mathop{\mathrm{Th}}(V) \wedge H} \\
    & {}
    \arrow["{z: \text{ forget support}}", from=1-2, to=1-5]
    \arrow[maps to, from=2-2, to=2-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMSwwLCJIX3tcXHRze2Y9MH19XntWXFxkdWFsfShYKSJdLFs0LDAsIkhee1ZcXGR1YWx9KFgpIl0sWzEsMl0sWzAsMCwiZV5IKFYsIGYpIFxcaW4iXSxbMSwxLCJYXFx0byB7WFxcb3ZlciBYXFxzbXRze2Y9MH19Il0sWzQsMSwiXFxUaChWKSBcXHNtYXNocHJvZCBIIl0sWzAsMSwiejogXFx0ZXh0eyBmb3JnZXQgc3VwcG9ydH0iXSxbNCw1LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV1d}{Link
to Diagram}
\end{quote}

Letting \(f: X\to \mathop{\mathrm{Th}}(V) \wedge H\), any two sections
\(f_1, f_2\) of \(V\) are connected by copies of \({\mathbb{A}}^1\) in
\(H^0(V)\), so
\begin{align*}
e^H(V) \coloneqq z(e^H(V, f_1)) = z(e^H(V, f_2))
.\end{align*}

\end{remark}

\begin{definition}[Euler number]

The \textbf{Euler number} \(n^H(V)\) of \(V\) in \(H^0(S)\) is
\begin{align*}
n^H(V) \coloneqq p_* e^H(V)
.\end{align*}

\end{definition}

\begin{remark}

This agrees with
\begin{align*}
n(V, f) = \sum_{x\in \left\{{f= 0 }\right\} \subseteq X} \deg_x f, && \deg_x f \in {\operatorname{GW}}(k)
,\end{align*}
for \(H = H\tilde{{\mathbb{Z}}}, {\operatorname{KO}}\) over
\(S\coloneqq\operatorname{Spec}k\). Moreover, \(n(V, f)\) is independent
of the choice of section.

\begin{quote}
See \href{https://arxiv.org/abs/1805.05920}{Déglise-Jin-Khan},
\href{https://arxiv.org/abs/2002.01848}{Bachmann-Wickelgren}.
\end{quote}

\end{remark}

\hypertarget{arithmetic-count-of-lines-on-a-smooth-cubic-surface-joint-with-jesse-kass}{%
\subsection{Arithmetic count of lines on a smooth cubic surface (joint
with Jesse
Kass)}\label{arithmetic-count-of-lines-on-a-smooth-cubic-surface-joint-with-jesse-kass}}

\begin{definition}[Cubic surface]

A \textbf{cubic surface} is given by \(X \subseteq {\mathbb{P}}^3\)
where \(X = \left\{{f=0}\right\}\) with \(f\in k[w,x,y,z]_3\)
homogeneous of degree 3.

\end{definition}

\begin{theorem}[Salmon-Cayley, 1849]

Any smooth cubic surface over \({\mathbb{C}}\) contains exactly 27
lines.

\end{theorem}

\begin{example}[?]

Consider the Fermat cubic:
\begin{align*}
f(x,y,z,w) = x^2 + y^2 + z^2 + w^2
.\end{align*}
The lines are given by
\begin{align*} 
\left\{{ {\left[ {S, -S, T, -T} \right]}  {~\mathrel{\Big|}~}[S, T] \in {\mathbb{P}}^1({\mathbb{C}}) \subseteq X }\right\}
.\end{align*}
For \(\omega^3,\lambda^3 = 1\), we can take
\begin{align*} 
\left\{{{\left[ {S, \lambda S, T, \omega T} \right]} {~\mathrel{\Big|}~}[S, T]\in {\mathbb{P}}^1({\mathbb{C}}) }\right\} 
.\end{align*}
Permuting the variables in \({4\choose 2}/2 =3\) ways, then there are
\(3\cdot 3 \cdot 3 = 27\) total lines.

\end{example}

\begin{proof}[?]

Let \({\operatorname{Gr}}(1, 3)\) be the Grassmannian parameterizing
\(W \subseteq {\mathbb{C}}^4\) with \(\dim W=2\), or equivalently copies
of \({\mathbb{P}}^1\) in \({\mathbb{P}}^3\). Take the tautological
\(S\to {\operatorname{Gr}}(1, 3)\) whose fiber over \([{\mathbb{P}}W]\)
is \(W\) itself. Then
\(\operatorname{Sym}^3 S {}^{ \vee }_{[{\mathbb{P}}W]} = \operatorname{Sym}^3(W {}^{ \vee })\)
are cubic polynomials in \(W\), and \(f\) determines a section
\(\sigma_f\) of \(\operatorname{Sym}^3 S {}^{ \vee }\) given by
\begin{align*}
\sigma_f([{\mathbb{P}}W) = { \left.{{f}} \right|_{{W}} }
,\end{align*}
which is zero iff the line \({\mathbb{P}}W\) is contained in \(X\). So
we've reduced to counting zeros of a section:
\begin{align*}
n(V) = \sum_{\text{lines } L \subseteq X} \deg_L \sigma_f
.\end{align*}

\begin{fact}

For a smooth cubic surface, all zeroes of \(\sigma_f\) have multiplicity
one.

\end{fact}

Over \({\mathbb{C}}\), the classical differential topological degree is
1, so \(n(V)\) is the number of lines, and \(n(V) = 27\).

Consider now the situation over \({\mathbb{R}}\):

\begin{itemize}
\tightlist
\item
  Schläfli (1861): Over \({\mathbb{R}}\), there can only be
  \(3, 7, 15\), or \(27\) lines.
\item
  Segre (1942): These lines break into two classes: hyperbolic, or
  elliptic.
\end{itemize}

For \(L \subseteq X\) a real line, \(L\) gives an involution
\(I:L\to L\) where \(I(p)\) is defined by consider \(T_p X\), which
contains a line \(L\) along with a curve \(C\) of degree 2 since the
total degree is 3:

\includegraphics{figures/2021-07-16_14-28-57.png}

In particular, there is another point in the intersection of
\(L \cap C \cap X\), so \(I(p)\) is defined as that point where
\(T_{I(p)}X = T_pX\). If \(I\) yields a
\({\mathbb{C}}{\hbox{-}}\)conjugate pair, say \(L\) is elliptic, and
otherwise if \(I\) yields two \({\mathbb{R}}{\hbox{-}}\)point say \(L\)
is hyperbolic. Taking a path \(p\to I(p)\), if the tangent space
undergoes a full twist, this is the elliptic situation.

\end{proof}

\begin{theorem}[Segre + many authors in the 2010s]

The number of hyperbolic lines minus the number of elliptic lines is
exactly 3.

\end{theorem}

\begin{question}

What about other fields, like
\(k = {\mathbb{F}}_p, { {\mathbb{Q}}_p }, {\mathbb{Q}}\)?

\end{question}

\begin{answer}

The above proof works in \({\mathbb{A}}^1{\hbox{-}}\)homotopy theory.
Letting \(X \subseteq X \subseteq {\mathbb{P}}^3_{/ {k}}\) a cubic
surface.

\end{answer}

\begin{definition}[Type of a line]

The \textbf{type} of a line \(L\) is an element
\begin{align*}
\left\langle{D}\right\rangle \in {\operatorname{GW}}(k(L)),
&&
D \in k(L)^{\times}/ (k(L)^{\times}){ {}^{ \scriptscriptstyle\times^{2} }  }
\end{align*}
such that the fixed points \(\mathrm{Fix} (I)\) form a conjugate pair of
points defined over \(k(L) { \left[ {\sqrt D} \right] }\).

\end{definition}

\begin{theorem}[Kass-Wickelgren]

let \(k\in \mathsf{Field}^{\operatorname{ch}\neq 2}\) and
\(X \subseteq {\mathbb{P}}^3_{/ {k}}\), then
\begin{align*}
\sum_{\substack{\text{lines} \\ L \subseteq k(X) }} \operatorname{Tr}_{k(L) / k} {\operatorname{type}}(D) = 15\left\langle{1}\right\rangle + 12\left\langle{-1}\right\rangle \in {\operatorname{GW}}(k)
.\end{align*}

\end{theorem}

\hypertarget{matthew-morrow-talk-1-thursday-july-15}{%
\section{Matthew Morrow, Talk 1 (Thursday, July
15)}\label{matthew-morrow-talk-1-thursday-july-15}}

\hypertarget{intro-7}{%
\subsection{Intro}\label{intro-7}}

\textbf{Abstract}:

\begin{quote}
Motivic cohomology offers, at least in certain situations, a geometric
refinement of algebraic K-theory or its variants (G-theory, KH-theory,
étale K-theory, \(\cdots\)). We will overview some aspects of the
subject, ranging from the original cycle complexes of Bloch, through
Voevodsky's work over fields, to more recent p-adic developments in the
arithmetic context where perfectoid and prismatic techniques appear.
\end{quote}

\textbf{References/Background}:

\begin{itemize}
\tightlist
\item
  Algebraic geometry, sheaf theory, cohomology.

  \begin{itemize}
  \tightlist
  \item
    Comfort with derived techniques such as descent and the cotangent
    complex would be helpful.
  \item
    Casual familiarity with K-theory, cyclic homology, and their
    variants would be motivational.
  \item
    Infinity-categories and spectra will appear, though probably not in
    a very essential way.
  \end{itemize}
\item
  \href{https://www.ias.edu/sites/default/files/Morrow\%20lectures\%201\%2B2.pdf}{Lecture
  Notes}
\end{itemize}

\begin{remark}

Some things we've already seen that will be useful:

\begin{itemize}
\tightlist
\item
  Motivic complexes
\item
  Milnor \({\mathsf{K}}{\hbox{-}}\)theory
\item
  Their relations to étale cohomology (e.g.~Bloch-Kato)
\item
  \({\mathbb{A}}^1{\hbox{-}}\)homotopy theory
\item
  Categorical aspects (e.g.~presheaves with transfer)
\end{itemize}

These have typically been for \({\mathsf{sm}}{\mathsf{Var}}_{/ {k}}\).
Our goals will be to study

\begin{itemize}
\tightlist
\item
  Motivic cohomology as a tool to analyze algebraic
  \({\mathsf{K}}{\hbox{-}}\)theory.
\item
  Recent progress in mixed characteristic, with fewer
  smoothness/regularity hypothesis
\end{itemize}

\end{remark}

\hypertarget{mathsfk_0-and-mathsfk_1}{%
\subsection{\texorpdfstring{\({\mathsf{K}}_0\) and
\({\mathsf{K}}_1\)}{\{\textbackslash mathsf\{K\}\}\_0 and \{\textbackslash mathsf\{K\}\}\_1}}\label{mathsfk_0-and-mathsfk_1}}

\begin{remark}

Some phenomena of \({\mathsf{K}}{\hbox{-}}\)theory to keep in mind:

\begin{itemize}
\tightlist
\item
  It encodes other invariants.
\item
  It breaks into ``simpler'' pieces that are motivic in nature.
\end{itemize}

\end{remark}

\begin{definition}[The Grothendieck group (Grothendieck, 50s)]

Let \(R\in \mathsf{CRing}\), then define the \textbf{Grothendieck group}
\({\mathsf{K}}_0(R)\) as the free abelian group:
\begin{align*}
{\mathsf{K}}_0(R) = {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, {\mathrm{fg}}, \cong} / \sim
.\end{align*}
where \([P] \sim [P'] + [P'']\) when there is a SES
\begin{align*}
0 \to P' \to P \to P'' \to 0
.\end{align*}

\end{definition}

\begin{remark}

There is an equivalent description as a group completion:
\begin{align*}
{\mathsf{K}}_0(R) = \qty{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, {\mathrm{fg}}, \cong}, \oplus }^ {\operatorname{gp} } 
.\end{align*}

The same definitions work for any \(X\in{\mathsf{Sch}}\) by replacing
\({\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, {\mathrm{fg}}}\)
with \({ { {\mathsf{Bun}}_{\operatorname{GL}_r} }}_{/ {X}}\), the
category of (algebraic) vector bundles over \(X\).

\end{remark}

\begin{example}[?]

For \(F\in\mathsf{Field}\), the dimension induces an isomorphism:
\begin{align*}
\dim_F: {\mathsf{K}}_0(F) &\to {\mathbb{Z}}\\
[P] &\mapsto \dim_F P
.\end{align*}

\end{example}

\begin{example}[?]

Let \({\mathcal{O}}\in\mathsf{DedekindDom}\), e.g.~the ring of integers
in a number field, then any ideal \(I{~\trianglelefteq~}{\mathcal{O}}\)
is a finite projective module and defines some
\([I] \in{\mathsf{K}}_0({\mathcal{O}})\). There is a SES
\begin{align*}
0 \to { \operatorname{Cl}} ({\mathcal{O}}) \xrightarrow{I \mapsto [I] - [{\mathcal{O}}] } {\mathsf{K}}_0({\mathcal{O}}) \xrightarrow{\operatorname{rank}_{\mathcal{O}}({-})} {\mathbb{Z}}\to 0
.\end{align*}
Thus \({\mathsf{K}}_0({\mathcal{O}})\) breaks up as
\({ \operatorname{Cl}} ({\mathcal{O}})\) and \({\mathbb{Z}}\), where the
class group is a classical invariant: isomorphism classes of nonzero
ideals.

\end{example}

\begin{example}[?]

Let
\(X\in{\mathsf{sm}}{\mathsf{Alg}}{\mathsf{Var}}^{{\mathrm{qproj}}}_{/ {k}}\)
over a field, and let \(Z\hookrightarrow X\) be an irreducible closed
subvariety. We can resolve the structure sheaf \({\mathcal{O}}_Z\) by
vector bundles:
\begin{align*}
0 \leftarrow{\mathcal{O}}_Z \leftarrow P_0 \leftarrow\cdots P_d \leftarrow 0
.\end{align*}
We can then define
\begin{align*}
[Z] \coloneqq\sum_{i=0}^d (-1)^i [P_i] \in{\mathsf{K}}_0(X)
,\end{align*}
which turns out to be independent of the resolution picked. This yields
a filtration:
\begin{align*}
{\operatorname{Fil}}_j{\mathsf{K}}_0(X) \coloneqq\left\langle{[Z] {~\mathrel{\Big|}~}Z\hookrightarrow X \text{ irreducible closed, } \operatorname{codim}(Z) \leq j}\right\rangle \\ \\ 
\implies{\mathsf{K}}_0(X) \supseteq{\operatorname{Fil}}_d{\mathsf{K}}_0(X) \supseteq\cdots \supseteq{\operatorname{Fil}}_0{\mathsf{K}}_0(X) \supseteq 0
.\end{align*}

\end{example}

\begin{theorem}[Part of Riemann-Roch]

There is a well-defined surjective map
\begin{align*}
{\operatorname{CH}}_j(X) \coloneqq\left\{{j{\hbox{-}}\text{dimensional cycles}}\right\} / \text{rational equivalence} &\to { {\operatorname{Fil}}_j{\mathsf{K}}_0(X) \over {\operatorname{Fil}}_{j-1}{\mathsf{K}}_0(X) } \\
Z &\mapsto [Z]
,\end{align*}
and the kernel is annihilated by \((j-1)!\).

\end{theorem}

\begin{slogan}

Up to small torsion, \({\mathsf{K}}_0(X)\) breaks into Chow groups.

\end{slogan}

\begin{definition}[Bass, 50s]

Set
\begin{align*}
{\mathsf{K}}_1(R)\coloneqq\operatorname{GL}(R)/E(R) \coloneqq\displaystyle\bigcup_{n\geq 1} \operatorname{GL}_n(R)/E_n(R)
\end{align*}
where we use the block inclusion
\begin{align*}
\operatorname{GL}_n(R) &\hookrightarrow\operatorname{GL}_{n+1} \\
g &\mapsto {
\begin{bmatrix}
  {g} & {0} 
\\
  {0} & {1}
\end{bmatrix}
}
\end{align*}
and \(E_n(R) \subseteq \operatorname{GL}_n(R)\) is the subgroup of
elementary row and column operations performed on \(I_n\).

\end{definition}

\begin{example}[?]

There exists a determinant map
\begin{align*}
\operatorname{det}: {\mathsf{K}}_1(R) &\to R^{\times}\\
g & \mapsto \operatorname{det}(g)
,\end{align*}
which has a right inverse
\(r\mapsto \operatorname{diag}(r,1,1,\cdots,1)\).

\end{example}

\begin{example}[?]

For \(F\in\mathsf{Field}\), we have
\(E_n(F) = {\operatorname{SL}}_n(F)\) by Gaussian elimination. Since
every \(g\in{\operatorname{SL}}_n(F)\) satisfies
\(\operatorname{det}(g) = 1\), there is an isomorphism
\begin{align*}
\operatorname{det}: {\mathsf{K}}_1(F) \xrightarrow{\sim} F^{\times}
.\end{align*}

\end{example}

\begin{remark}

We can see a relation to étale cohomology here by using Kummer theory to
identify
\begin{align*}
{\mathsf{K}}_1(F) / m \xrightarrow{\sim} F^{\times}/m \xrightarrow{\text{Kummer}, \sim} H^1_{ \mathsf{Gal}} (F; \mu_m)
\end{align*}
for \(m\) prime to \(\operatorname{ch}F\), so this is an easy case of
Bloch-Kato.

\end{remark}

\begin{example}[?]

For \({\mathcal{O}}\) the ring of integers in a number field, there is
an isomorphism
\begin{align*}
\operatorname{det}: {\mathsf{K}}_1({\mathcal{O}}) \xrightarrow{\sim} {\mathcal{O}}^{\times}
,\end{align*}
but this is now a deep theorem due to Bass-Milnor-Serre, Kazhdan.

\end{example}

\begin{example}[?]

Let
\(D \coloneqq{\mathbb{R}}[x, y] / \left\langle{x^2 + y^2 - 1}\right\rangle \in\mathsf{DedekindDom}\),
then there is a nonzero class
\begin{align*}
{
\begin{bmatrix}
  {x} & {y} 
\\
  {-y} & {x}
\end{bmatrix}
} \in \ker \operatorname{det}
,\end{align*}
so the previous result for \({\mathcal{O}}\) is not a general fact about
Dedekind domains. It turns out that
\begin{align*}
{\mathsf{K}}_1(D) \xrightarrow{\sim} D^{\times}\oplus {\mathcal{L}}
,\end{align*}
where \({\mathcal{L}}\) encodes some information about loops which
vanishes for number fields.

\end{example}

\hypertarget{higher-algebraic-mathsfkhbox-theory}{%
\subsection{\texorpdfstring{Higher Algebraic
\({\mathsf{K}}{\hbox{-}}\)theory}{Higher Algebraic \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory}}\label{higher-algebraic-mathsfkhbox-theory}}

\begin{remark}

By the 60s, it became clear that \({\mathsf{K}}_0, {\mathsf{K}}_1\)
should be the first graded pieces in some exceptional cohomology theory,
and there should exist some \({\mathsf{K}}_n(R)\) for all \(n\geq 0\)
(to be defined). Quillen's Fields was a result of proposing multiple
definitions, including the following:

\end{remark}

\begin{definition}[The $\K\dash$theory spectrum (Quillen, 73)]

Define a \({\mathsf{K}}{\hbox{-}}\)theory space or spectrum (infinite
loop space) by deriving the functor \({\mathsf{K}}_0({-})\):
\begin{align*}
K(R) \coloneqq \mathsf{B}\mkern-3mu \operatorname{GL} (R){ {}^{+} }\times{\mathsf{K}}_0(R)
\end{align*}
where
\(\pi_* \mathsf{B}\mkern-3mu \operatorname{GL} (R) = \operatorname{GL}(R)\)
for \(*=1\). Quillen's plus construction forces \(\pi_*\) to be abelian
without changing the homology, although this changes homotopy in higher
degrees. We then define
\begin{align*}
{\mathsf{K}}_n(R) \coloneqq\pi_n {\mathsf{K}}(R)
.\end{align*}

\end{definition}

\begin{remark}

This construction is good for the (hard!) hands-on calculations Quillen
originally did, but a more modern point of view would be

\begin{itemize}
\tightlist
\item
  Setting \({\mathsf{K}}(R)\) to be the \(\infty{\hbox{-}}\)group
  completion of the \({\mathbb{E}}_\infty\) space associated to the
  category
  \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, \cong}\).
\item
  Regarding \({\mathsf{K}}({-})\) as the universal invariant of
  \(\mathsf{Stab}{ \underset{\infty}{ \mathsf{Cat}} }\) taking exact
  sequences in
  \({\operatorname{Stab}}{ \underset{\infty}{ \mathsf{Cat}} }\) to
  cofibers sequences in the category of spectra \({\mathsf{Sp}}\), in
  which case one defines
  \begin{align*}
  {\mathsf{K}}(R) \coloneqq{\mathsf{K}}(\mathsf{Perf}\mathsf{Ch}\qty{ {\mathsf{R}{\hbox{-}}\mathsf{Mod}} }  ) 
  \end{align*}
  as \({\mathsf{K}}({-})\) of perfect complexes of
  \(R{\hbox{-}}\)modules.
\end{itemize}

Both constructions output groups \({\mathsf{K}}_n(R)\) for \(n\geq 0\).

\end{remark}

\begin{example}[Quillen, 73]

The only complete calculation of \(K\) groups that we have is
\begin{align*}
{\mathsf{K}}_n({\mathbb{F}}_q) = 
\begin{cases}
{\mathbb{Z}}& n=0 
\\
0 & n \text{ even}
\\
{\mathbb{Z}}/\left\langle{q^{ {n+1\over 2} - 1 }}\right\rangle & n \text{ odd}.
\end{cases}
\end{align*}

\end{example}

\begin{example}[?]

We know \({\mathsf{K}}\) groups are hard because
\({\mathsf{K}}_{n>0}({\mathbb{Z}}) = 0 \iff\) the Vandiver conjecture
holds, which is widely open.

\todo[inline]{Check content of conjecture, maybe 4n?}

\end{example}

\begin{conjecture}

If
\(R \in {\mathsf{Alg}}_{/ {{\mathbb{Z}}}} ^{{\mathrm{ft}}, \mathrm{reg}}\)
then \({\mathsf{K}}_n(R)\) should be a finitely generated abelian group
for all \(n\). This is widely open, but known when \(\dim R \leq 1\).

\end{conjecture}

\begin{example}[?]

For \(F\in\mathsf{Field}\) with \(\operatorname{ch}F\) prime to
\(m\geq 1\), ten
\begin{align*}
\operatorname{TateSymb}: {\mathsf{K}}_2(F) / m \xrightarrow{\sim} H^2_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})
,\end{align*}
which is a specialization of Bloch-Kato due to Merkurjev-Suslin.

\end{example}

\begin{example}[Lichtenbaum, Quillen 70s]

Partially motivated by special values of zeta functions, for a number
field \(F\) and \(m\geq 1\), formulae for
\({\mathsf{K}}_n(F; {\mathbb{Z}}/m)\) were conjectured in terms of
\(H_\text{ét}\).

\end{example}

\begin{remark}

Here we're using \textbf{\({\mathsf{K}}{\hbox{-}}\)theory with
coefficients}, where one takes a spectrum and constructs a mod \(m\)
version of it fitting into a SES
\begin{align*}
0\to {\mathsf{K}}_n(F)/m \to {\mathsf{K}}_n(F; {\mathbb{Z}}/m) \to {\mathsf{K}}_{n-1}(F)[m] \to 0
.\end{align*}
However, it can be hard to reconstruct \({\mathsf{K}}_n({-})\) from
\({\mathsf{K}}_n({-}, {\mathbb{Z}}/m)\).

\end{remark}

\hypertarget{arrival-of-motivic-cohomology}{%
\subsection{Arrival of Motivic
Cohomology}\label{arrival-of-motivic-cohomology}}

\begin{question}

\({\mathsf{K}}{\hbox{-}}\)theory admits a refinement in the form of
motivic cohomology, which splits into simpler pieces such as étale
cohomology. In what generality does this phenomenon occur?

\end{question}

\begin{example}[?]

This is always true in topology: given \(X\in {\mathsf{Top}}\),
\({\mathsf{K}}_0^{\mathsf{Top}}\) can be defined using complex vector
bundles, and using suspension and Bott periodicity one can define
\({\mathsf{K}}_n^{\mathsf{Top}}(X)\) for all \(n\).

\end{example}

\begin{theorem}[Atiyah-Hirzebruch]

There is a spectral sequence which degenerates rationally:
\begin{align*}
E_2^{i,j} = H_{\operatorname{Sing}}^{i-j}(X; {\mathbb{Z}}) \Rightarrow{\mathsf{K}}^{\mathsf{Top}}_{-i-j}(X)
.\end{align*}

\end{theorem}

\begin{remark}

So up to small torsion, topological \({\mathsf{K}}{\hbox{-}}\)theory
breaks up into singular cohomology. Motivated by this, we have the
following

\end{remark}

\hypertarget{big-conjecture}{%
\subsection{Big Conjecture}\label{big-conjecture}}

\begin{conjecture}[Existence of motivic cohomology (Beilinson-Lichtenbaum, 80s)]

For any \(X\in{\mathsf{sm}}{\mathsf{Var}}_{/ {k}}\), there should exist
\textbf{motivic complexes}
\begin{align*}
{\mathbb{Z}}_{ \mathrm{mot}} (j)(X), && j\geq 0
\end{align*}
whose homology, the \textbf{weight \(j\) motivic cohomology of \(X\)},
has the following expected properties:

\begin{itemize}
\item
  There is some analog of the Atiyah-Hirzebruch spectral sequence which
  degenerates rationally:
  \begin{align*}
  E_2^{i, j} = H_{ \mathrm{mot}} ^{i-j}(X; {\mathbb{Z}}(-j)) \Rightarrow{\mathsf{K}}_{-i-j}(X)
  ,\end{align*}
  where \(H_{ \mathrm{mot}} ^*({-})\) is taking kernels mod images for
  the complex \({\mathbb{Z}}_{ \mathrm{mot}} (\bullet)(X)\) satisfying
  descent.
\item
  In low weights, we have

  \begin{itemize}
  \tightlist
  \item
    \({\mathbb{Z}}_{ \mathrm{mot}} (0)(X) = {\mathbb{Z}}^{\# \pi_0(X)}[0]\)
    in degree 0, supported in degree zero.
  \item
    \({\mathbb{Z}}_{ \mathrm{mot}} (1)(X) = {\mathbb{R}}\Gamma_{\mathrm{zar}}(X; {\mathcal{O}}_X^{\times})[-1]\),
    supported in degrees 1 and 2 for a normal scheme after the
    right-shift.
  \end{itemize}
\item
  Range of support: \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)\) is supported
  in degrees \(0,\cdots, 2j\), and in degrees \(\leq j\) if
  \(X=\operatorname{Spec}R\) for \(R\) a local ring.
\item
  Relation to Chow groups:
  \begin{align*}
  H^{2j}_{ \mathrm{mot}} (X; {\mathbb{Z}}(j)) { { \, \xrightarrow{\sim}\, }}{\operatorname{CH}}^j(X)
  .\end{align*}
\item
  Relation to étale cohomology (Beilinson-Lichtenbaum conjecture):
  taking the complex mod \(m\) and taking homology yields
  \begin{align*}
  H_{ \mathrm{mot}} ^i(X; {\mathbb{Z}}/m(j)) \xrightarrow{\sim} H^i_\text{ét}(X; \mu_m^{\otimes j})
  \end{align*}
  if \(m\) is prime to \(\operatorname{ch}k\) and \(i\leq j\).
\end{itemize}

\end{conjecture}

\begin{example}[?]

Considering computing \({\mathsf{K}}_n(F) \pmod m\) for \(m\) odd and
for number fields \(F\), as predicted by Lichtenbaum-Quillen. The mod
\(m\) AHSS is simple in this case, since
\(\operatorname{cohdim}F \leq 2\):

\begin{center}
\begin{tikzcd}
    \bullet & \bullet & \bullet & \bullet \\
    \bullet & \bullet & \bullet & {H_{ \mathsf{Gal}} ^0(F; {\mathbb{Z}}/m)} \\
    \bullet & \bullet & {H^0_{ \mathsf{Gal}} (F; \mu_m)} & {H_{ \mathsf{Gal}} ^1(F; \mu_m)} \\
    \bullet & {H^0_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} & {H^1_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} & {H_{ \mathsf{Gal}} ^2(F; \mu_m^{\otimes 2})} \\
    \vdots & \vdots & {H^2_{ \mathsf{Gal}} (F; \mu_m^{\otimes 3})} & \bullet \\
    \vdots & \vdots & \bullet & \vdots
    \arrow["\partial", from=4-2, to=5-4]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

The differentials are all zero, so we obtain
\begin{align*}
{\mathsf{K}}_{2j-1}(F; {\mathbb{Z}}/m) \xrightarrow{\sim} H^1_{ \mathsf{Gal}} (F; \mu_m^{\otimes j})
\end{align*}
and
\begin{align*}
0 \to H^2_{ \mathsf{Gal}} (F, \mu_m^{\otimes j+1}) \to {\mathsf{K}}_{2j}(F; {\mathbb{Z}}/m) \to H_{{ \mathsf{Gal}} }^0(F; \mu_m^{\otimes j})\to 0
.\end{align*}

\end{example}

\begin{theorem}[Bloch, Levine, Friedlander, Rost, Suslin, Voevodsky, $\cdots$]

The above conjectures are true \textbf{except} for Beilinson-Soulé
vanishing, i.e.~the conjecture that
\({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)\) is supported in positive degrees
\(n\geq 0\).

\end{theorem}

\begin{remark}

Remarkably, one can write a definition somewhat easily which turns out
to work in a fair amount of generality for schemes over a Dedekind
domain.

\end{remark}

\begin{definition}[Higher Chow groups]

For \(X\in {\mathsf{Var}}_{/ {k}}\), let \(z^j(X, n)\) be the free
abelian group of codimension \(j\) irreducible closed subschemes of
\(X { \underset{\scriptscriptstyle {F} }{\times} } \Delta^n\)
intersecting all faces properly, where
\begin{align*}
\Delta^n = \operatorname{Spec}\qty{F[T_0, \cdots, T_n] \over \left\langle{\sum T_i - 1}\right\rangle} \cong {\mathbb{A}}^n_{/ {F}} 
,\end{align*}
which contains ``faces'' \(\Delta^m\) for \(m\leq n\), and
\emph{properly} means the intersections are of the expected codimension.
Then \textbf{Bloch's complex of higher cycles} is the complex
\(z^j(X, \bullet)\) where the boundary map is the alternating sum
\begin{align*}
z^j(X, n) \ni {{\partial}}(Z) = \sum_{i=0}^n (-1)^i [Z \cap\mathrm{Face}_i(X\times \Delta^{n-1})]
,\end{align*}
\textbf{Bloch's higher Chow groups} are the cohomology of this complex:
\begin{align*}
\mathsf{Ch}^j(X, n) \coloneqq H_n(z^j(X, \bullet))
,\end{align*}
and then the following complex has the expected properties:
\begin{align*}
{\mathbb{Z}}_{ \mathrm{mot}} (j)(X) \coloneqq z^j(X, \bullet)[-2j]
\end{align*}

\end{definition}

\begin{remark}

Déglise's talks present the machinery one needs to go through to verify
this!

\end{remark}

\hypertarget{milnor-mathsfkhbox-theory-and-bloch-kato}{%
\subsection{\texorpdfstring{Milnor \({\mathsf{K}}{\hbox{-}}\)theory and
Bloch-Kato}{Milnor \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory and Bloch-Kato}}\label{milnor-mathsfkhbox-theory-and-bloch-kato}}

\begin{remark}

How is motivic cohomology related to the Bloch-Kato conjecture? Recall
from Danny's talks that for \(F\in\mathsf{Field}\) then one can form
\begin{align*}
 {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) = (F^{\times}){ {}^{ \scriptstyle\otimes_{F}^{j} }  } / \left\langle{\text{Steinberg relations}}\right\rangle
,\end{align*}
and for \(m\geq 1\) prime to \(\operatorname{ch}F\) we can take
Tate/Galois/cohomological symbols
\begin{align*}
\operatorname{TateSymb}:  {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/m \to H^j_{ \mathsf{Gal}} (F; \mu_m^{\otimes j})
.\end{align*}
where \(\mu_m^{\otimes j}\) is the \(j\)th Tate twist. Bloch-Kato
conjectures that this is an isomorphism, and it is a theorem due to
Rost-Voevodsky that the Tate symbol is an isomorphism. The following
theorem says that a piece of \(H_{ \mathrm{mot}}\) can be identified as
something coming from \({\mathsf{K}}^{\scriptstyle\mathrm{M}}\):

\end{remark}

\begin{theorem}[Nesterenko-Suslin, Totaro]

For any \(F\in\mathsf{Field}\), for each \(j\geq 1\) there is a natural
isomorphism
\begin{align*}
 {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) \xrightarrow{\sim} H_{ \mathrm{mot}} ^j(F; {\mathbb{Z}}(j))
.\end{align*}

\end{theorem}

\begin{remark}

Taking things mod \(m\) yields
\begin{align*}
 {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/m \xrightarrow{\sim} H_{ \mathrm{mot}} ^j(F; {\mathbb{Z}}/m(j))
\xrightarrow{\sim, \text{BL}} H_\text{ét}^j(F; \mu_m^{\otimes j})
,\end{align*}
where the conjecture is that the obstruction term for the first
isomorphism coming from \(H^{j+1}\) vanishes for local objects, and
Beilinson-Lichtenbaum supplies the second isomorphism. The composite is
the Bloch-Kato isomorphism, so Beilinson-Lichtenbaum \(\implies\)
Bloch-Kato, and it turns out that the converse is essentially true as
well. This is also intertwined with the Hilbert 90 conjecture.

Tomorrow: we'll discard coprime hypotheses, look at \(p{\hbox{-}}\)adic
phenomena, and look at what happens étale locally.

\end{remark}

\hypertarget{matthew-morrow-talk-2-friday-july-16}{%
\section{Matthew Morrow, Talk 2 (Friday, July
16)}\label{matthew-morrow-talk-2-friday-july-16}}

\begin{remark}

A review of yesterday:

\begin{itemize}
\item
  \({\mathsf{K}}{\hbox{-}}\)theory can be refined by motivic cohomology,
  i.e.~it breaks into pieces. More precisely we have the
  Atiyah-Hirzebruch spectral sequence, and even better, the spectrum
  \({\mathsf{K}}(X)\) has a motivic filtration with graded pieces
  \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)[2j]\).
\item
  The \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)\) correspond to algebraic
  cycles and étale cohomology mod \(m\), where \(m\) is prime to
  \(\operatorname{ch}k\), due to Beilinson-Lichtenbaum and
  Beilinson-Bloch.
\end{itemize}

Today we'll look at the classical mod \(p\) theory, and variations on a
theme: e.g.~replacing \({\mathsf{K}}{\hbox{-}}\)theory with similar
invariants, or weakening the hypotheses on \(X\). We'll also discuss
recent progress in the case of étale \({\mathsf{K}}{\hbox{-}}\)theory,
particularly \(p{\hbox{-}}\)adically.

\end{remark}

\hypertarget{mod-p-motivic-cohomology-in-characteristic-p}{%
\subsection{\texorpdfstring{Mod \(p\) motivic cohomology in
characteristic
\(p\)}{Mod p motivic cohomology in characteristic p}}\label{mod-p-motivic-cohomology-in-characteristic-p}}

\begin{remark}

For \(F\in\mathsf{Field}\) and \(m\geq 1\) prime to
\(\operatorname{ch}F\), the Atiyah-Hirzebruch spectral sequence mod
\(m\) takes the following form:
\begin{align*}
E_2^{i, j} = H_{ \mathrm{mot}} ^{i, j}(F, {\mathbb{Z}}/m(-j)) 
\overset{BL}{=}
\begin{cases}
 H^{i-j}_{ \mathsf{Gal}} (F; \mu_m^{\otimes j}) & i\leq 0  
\\
 0 & i>0 .
\end{cases}
.\end{align*}

Thus \(E_2\) is supported in a quadrant four wedge:

\includegraphics{figures/2021-07-16_11-12-08.png}

We know the axis:
\begin{align*}
H^j(F; \mu_m^{\otimes j}) \xrightarrow{\sim}  {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/m
.\end{align*}

What happens if \(m>p = \operatorname{ch}F\) for
\(\operatorname{ch}F > 0\)?

\end{remark}

\begin{theorem}[Izhbolidin (90), Bloch-Kato-Gabber (86), Geisser-Levine (2000)]

Let \(F\in \mathsf{Field}^{\operatorname{ch}= p}\), then

\begin{itemize}
\item
  \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)\) and
  \({\mathsf{K}}_j(F)\) are \(p{\hbox{-}}\)torsionfree.
\item
  \({\mathsf{K}}_j(F)/p \xhookleftarrow{} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/p \xhookrightarrow{\operatorname{dLog}} \Omega_F^j\)
\end{itemize}

\end{theorem}

\begin{definition}[$\dlog$]

The \(\operatorname{dLog}\) map is defined as
\begin{align*}
\operatorname{dLog}:  {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) / p &\to \Omega_f^j \\
\bigotimes_{i} \alpha_i &\mapsto \bigwedge\nolimits_i  {d \alpha_i \over \alpha_i}
,\end{align*}
and we write
\(\Omega^j_{F, \log} \coloneqq\operatorname{im}\operatorname{dLog}\).

\end{definition}

\begin{remark}

So the above theorem is about showing the injectivity of
\(\operatorname{dLog}\). What Geisser-Levine really prove is that
\begin{align*}
{\mathbb{Z}}_{ \mathrm{mot}} (j)(F)/p \xrightarrow{\sim} \Omega_{F, \log}^j[-j]
.\end{align*}
Thus the mod \(p\) Atiyah-Hirzebruch spectral sequence, just motivic
cohomology lives along the axis
\begin{align*}
E_2^{i, j} = 
\begin{cases}
\Omega_{F, \log}^{-j}  &  i=0
\\
0 & \text{else }
\end{cases}
\Rightarrow{\mathsf{K}}_{i-j}(F; {\mathbb{Z}}/p)
\end{align*}
and \({\mathsf{K}}_j(F)/p \xrightarrow{\sim} \Omega_{F, \log}^j\).

\end{remark}

\begin{remark}

So life is much nicer in \(p\) matching the characteristic! Some
remarks:

\begin{itemize}
\tightlist
\item
  The isomorphism remains true with \(F\) replaced any
  \(F\in {\mathsf{Alg}}_{/ {{\mathbb{F}}_p}} ^{\mathrm{reg}, {\mathsf{loc}}, \mathrm{Noeth}}\):
  \begin{align*}
  {\mathsf{K}}_j(F)/p \xrightarrow{\sim} \Omega_{F, \log}^j
  .\end{align*}
\item
  The hard part of the theorem is showing that mod \(p\), there is a
  surjection
  \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) \twoheadrightarrow{\mathsf{K}}_j(F)\).
  The proof goes through using \(z^j(F, \bullet)\) and the
  Atiyah-Hirzebruch spectral sequence, and seems to necessarily go
  through motivic cohomology.
\end{itemize}

\end{remark}

\begin{question}

Is there a direct proof? Or can one even just show that
\begin{align*}
{\mathsf{K}}_j(F)/p = 0 \text{ for } j> [F: {\mathbb{F}}_p]_{\mathrm{tr}}
?\end{align*}

\end{question}

\begin{conjecture}[Beilinson]

This becomes an isomorphism after tensoring to \({\mathbb{Q}}\), so
\begin{align*}
 {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) \otimes_{\mathbb{Z}}{\mathbb{Q}}\xrightarrow{\sim}  {\mathsf{K}}_j(F)\otimes_{\mathbb{Z}}{\mathbb{Q}}
.\end{align*}
This is known to be true for finite fields.

\end{conjecture}

\begin{conjecture}

\begin{align*}
H_{ \mathrm{mot}} ^i(F; Z(j)) \text{ is torsion unless }i=j
.\end{align*}
This is wide open, and would follow from the following:

\end{conjecture}

\begin{conjecture}[Parshin]

If \(X\in {\mathsf{sm}}{\mathsf{Var}}^{\mathop{\mathrm{proj}}}_{/ {k}}\)
over \(k\) a finite field, then
\begin{align*}
H_{ \mathrm{mot}} ^i(X; Z(j)) \text{ is torsion unless } i=2j
.\end{align*}

\end{conjecture}

\hypertarget{variants-on-a-theme}{%
\subsection{Variants on a theme}\label{variants-on-a-theme}}

\begin{question}

What things (other than \({\mathsf{K}}{\hbox{-}}\)theory) can be
motivically refined?

\end{question}

\hypertarget{mathsfghbox-theory}{%
\subsubsection{\texorpdfstring{\({\mathsf{G}}{\hbox{-}}\)theory}{\{\textbackslash mathsf\{G\}\}\{\textbackslash hbox\{-\}\}theory}}\label{mathsfghbox-theory}}

\begin{remark}

Bloch's complex \(z^j(X, \bullet)\) makes sense for any
\(X\in {\mathsf{Sch}}\), and for \(X\) finite type over \(R\) a field or
a Dedekind domain. Its homology yields an Atiyah-Hirzebruch spectral
sequence
\begin{align*}
E_2^{i, j} = {\operatorname{CH}}^{-j}(X, -i-j) \Rightarrow{\mathsf{G}}_{-i-j}(X)
,\end{align*}
where \({\mathsf{G}}{\hbox{-}}\)theory is the
\({\mathsf{K}}{\hbox{-}}\)theory of \({\mathsf{Coh}}(X)\). See Levine's
work.

Then \(z^j(X, \bullet)\) defines \textbf{motivic Borel-Moore
homology}\footnote{Note that this is homology and not cohomology!} which
refines \({\mathsf{G}}{\hbox{-}}\)theory.

\end{remark}

\hypertarget{mathsfkscriptscriptstyle-mathrmh-hbox-theory}{%
\subsubsection{\texorpdfstring{\({\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} {\hbox{-}}\)theory}{\{\textbackslash mathsf\{K\}\}\^{}\{\textbackslash scriptscriptstyle \textbackslash mathrm\{H\}\} \{\textbackslash hbox\{-\}\}theory}}\label{mathsfkscriptscriptstyle-mathrmh-hbox-theory}}

\begin{remark}

This is Weibel's ``homotopy invariant
\({\mathsf{K}}{\hbox{-}}\)theory'', obtained by forcing homotopy
invariance in a universal way, which satisfies
\begin{align*}
 {\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} (R[T]) \xrightarrow{\sim}  {\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} (R) && \forall R
.\end{align*}
One defines this as a simplicial spectrum
\begin{align*}
 {\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} (R) \coloneqq{ {\left\lvert {
q \mapsto {\mathsf{K}}\qty{R[T_0, \cdots, T_q] \over 1 - \sum_{i=0}^q T_i}
} \right\rvert} }
.\end{align*}

\end{remark}

\begin{remark}

One hopes that for (reasonable) schemes \(X\), there should exist an
\({\mathbb{A}}^1{\hbox{-}}\)invariant motivic cohomology such that

\begin{itemize}
\tightlist
\item
  There is an Atiyah-Hirzebruch spectral sequence converging to
  \({\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} _{i-j}(X)\).
\item
  Some Beilinson-Lichtenbaum properties.
\item
  Some relation to cycles.
\end{itemize}

For \(X\) Noetherian with \(\operatorname{krulldim}X<\infty\), the
state-of-the-art is that stable homotopy machinery can produce an
Atiyah-Hirzebruch spectral sequence using representability of
\({\mathsf{K}}^{\scriptscriptstyle \mathrm{H}}\) in \({\mathsf{SH}}(X)\)
along with the slice filtration.

\end{remark}

\hypertarget{motivic-cohomology-with-modulus}{%
\subsubsection{Motivic cohomology with
modulus}\label{motivic-cohomology-with-modulus}}

\begin{remark}

Let \(X\in{\mathsf{sm}}{\mathsf{Var}}\) and \(D\hookrightarrow X\) an
effective (not necessarily reduced) Cartier divisor -- thought of where
\(X\setminus D\) is an open which is compactified after adding \(D\).
Then one constructs \(z^j\qty{ {X\vert D }, \bullet}\) which are
complexes of cycles in ``good position'' with respect to the boundary
\(D\).

\end{remark}

\begin{conjecture}

There is an Atiyah-Hirzebruch spectral sequence
\begin{align*}
E_2^{i, j} = {\operatorname{CH}}^{j}\qty{ {X  \vert D }, (-i-j) } \Rightarrow{\mathsf{K}}_{-i-j}(X, D)
,\end{align*}
where the limiting term involves \emph{relative \(K{\hbox{-}}\)groups}.
So there is a motivic (i.e.~cycle-theoretic) description of relative
\({\mathsf{K}}{\hbox{-}}\)theory.

\end{conjecture}

\hypertarget{uxe9tale-mathsfkhbox-theory}{%
\subsection{\texorpdfstring{Étale
\({\mathsf{K}}{\hbox{-}}\)theory}{Étale \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory}}\label{uxe9tale-mathsfkhbox-theory}}

\begin{remark}

\({\mathsf{K}}{\hbox{-}}\)theory is simple étale-locally, at least away
from the residue characteristic.

\end{remark}

\begin{theorem}[Gabber, Suslin]

If \(A \in{\mathsf{loc}}\mathsf{Ring}\) is strictly Henselian with
residue field \(k\) and \(m \geq 1\) is prime to \(\operatorname{ch}k\),
then
\begin{align*}
{\mathsf{K}}_n(A; {\mathbb{Z}}/m) \xrightarrow{\sim} {\mathsf{K}}_n(k; {\mathbb{Z}}/m)
\xrightarrow{\sim} 
\begin{cases}
\mu_m(k)^{\otimes{n\over 2}} & n \text{ even}  
\\
0 & n \text{ odd}.
\end{cases}
\end{align*}

\end{theorem}

\begin{remark}

The problem is that \({\mathsf{K}}{\hbox{-}}\)theory does \emph{not}
satisfy étale descent!
\begin{align*}
\text{For } B\in{ \mathsf{Gal}} \mathsf{Field}_{/ {A}} ^{\deg < \infty},
&&
K(B)^{h{ \mathsf{Gal}} \qty{B_{/ {A}} }} \not\cong K(A)
.\end{align*}

View \({\mathsf{K}}{\hbox{-}}\)theory as a presheaf of spectra (in the
sense of infinity sheaves), and define \textbf{étale
\({\mathsf{K}}{\hbox{-}}\)theory} \(K^\text{ét}\) to be the universal
modification of \({\mathsf{K}}{\hbox{-}}\)theory to satisfy étale
descent. This was considered by Thomason, Soulé, Friedlander.

\end{remark}

\begin{remark}

Even better than \({\mathsf{K}}^\text{ét}\) is Clausen's \textbf{Selmer
\({\mathsf{K}}{\hbox{-}}\)theory}, which does the right thing
integrally. Up to subtle convergence issues, for any
\(X\in {\mathsf{Sch}}\) and \(m\) prime to \(\operatorname{ch}X\) (the
characteristic of the residue field) one gets an Atiyah-Hirzebruch
spectral sequence
\begin{align*}
E_2^{i, j} = H_\text{ét}^{i-j}(X; \mu_m^{\otimes-j}) \Rightarrow{\mathsf{K}}_{i-j}^{\text{ét}}(X; {\mathbb{Z}}/m)
.\end{align*}

Letting \(F\) be a field and \(m\) prime to \(\operatorname{ch}F\), the
spectral sequence looks as follows:

\begin{center}
\begin{tikzcd}
    &&&&&& {} \\
    \\
    \\
    \\
    \bullet &&&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0_{ \mathsf{Gal}} (F; {\mathbb{Z}}/m)} & {H^1(F; {\mathbb{Z}}/m)} &&&&&&& \bullet \\
    &&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0(F; \mu_m^{\otimes 1})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^1_{ \mathsf{Gal}} (F; \mu_m^{})} & {H^2(F; \mu_m^{})} \\
    &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0(F; \mu_m^{\otimes 2})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^1(F; \mu_m^{\otimes 2})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^2_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} & {H^3_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} \\
    &&& {} &&& \vdots \\
    &&&&&& {} \\
    &&&&&& {}
    \arrow[color={rgb,255:red,135;green,135;blue,135}, dotted, from=5-1, to=5-15]
    \arrow[color={rgb,255:red,135;green,135;blue,135}, dotted, from=1-7, to=10-7]
    \arrow[dashed, no head, from=5-7, to=8-4]
    \arrow[dashed, no head, from=5-7, to=9-7]
    \arrow[from=6-6, to=7-8]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

The whole thing converges to
\({\mathsf{K}}_{-i-j}^\text{ét}(F; {\mathbb{Z}}/m)\), and the sector
conjecturally converges to \({\mathsf{K}}_{-i-j}(F; {\mathbb{Z}}/m)\) by
the Beilinson-Lichtenbaum conjecture.

\end{remark}

\hypertarget{recent-progress}{%
\subsection{Recent Progress}\label{recent-progress}}

\begin{remark}

We now focus on

\begin{itemize}
\tightlist
\item
  Étale \({\mathsf{K}}{\hbox{-}}\)theory, \({\mathsf{K}}^\text{ét}\)
\item
  mod \(p\) coefficients, even period
\item
  \(p{\hbox{-}}\)adically complete rings
\end{itemize}

The last is not a major restriction, since there is an arithmetic gluing
square

\begin{center}
\begin{tikzcd}
    R && {R{ \left[ { \scriptstyle \frac{1}{p} } \right] }} \\
    \\
    {\widehat{R}} && {\widehat{R}{ \left[ { \scriptstyle \frac{1}{p} } \right] }}
    \arrow[from=1-1, to=3-1]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJSIl0sWzIsMCwiUlxcaW52ZXJ0e3B9Il0sWzIsMiwiXFxoYXR7Un1cXGludmVydHtwfSJdLFswLDIsIlxcaGF0e1J9Il0sWzAsM10sWzMsMl0sWzEsMl0sWzAsMV1d}{Link
to Diagram}
\end{quote}

Here the bottom-left is the \(p{\hbox{-}}\)adic completion, and the
right-hand side uses classical results when \(p\) is prime to all
residue characteristic classes.

\end{remark}

\begin{theorem}[Bhatt-M-Scholze, Antieau-Matthew-M-Nikolaus, Lüders–M, Kelly-M]

For any \(p{\hbox{-}}\)adically complete ring \(R\) (or in more
generality, derived \(p{\hbox{-}}\)complete simplicial rings) one can
associate a theory of \textbf{\(p{\hbox{-}}\)adic étale motivic
cohomology} -- \(p{\hbox{-}}\)complete complexes
\({\mathbb{Z}}_p(j)(R)\) for \(j\geq 0\) satisfying an analog of the
Beilinson-Lichtenbaum conjectures:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  An Atiyah-Hirzebruch spectral sequence:
  \begin{align*}
  E_2^{i, j} = H^{i-j}({\mathbb{Z}}_p(j)(R)) \Rightarrow{\mathsf{K}}_{-i-j}^\text{ét}(R; {\mathbb{Z}}){ {}_{ \widehat{p} } }
  .\end{align*}
\item
  Known low weights:
  \begin{align*}
  {\mathbb{Z}}_p(0)(R) &\xrightarrow{\sim} {\mathbb{R}}\Gamma_\text{ét}(R; {\mathbb{Z}}_p) \\
  {\mathbb{Z}}_p(1)(R) &\xrightarrow{\sim} { \overbrace{{\mathbb{R}}\Gamma_\text{ét}(R; {\mathbb{G}}_m)}^{\widehat{\hspace{4em}}}  }  [-1] 
  .\end{align*}
\item
  Range of support: \({\mathbb{Z}}_p(j)(R)\) is supported in degrees
  \(d\leq j+1\), and even in degrees \(d\leq n+1\) if the
  \(R{\hbox{-}}\)module \(\Omega_{R/pR}^1\) is generated by \(n'<n\)
  elements. It is supported in non-negative degrees if \(R\) is
  \textbf{quasisyntomic}, which is a mild smoothness condition that
  holds in particular if \(R\) is regular.
\item
  An analog of Nesterenko-Suslin: for
  \(R \in {\mathsf{loc}}\mathsf{Ring}\),
  \begin{align*}
  { \widehat{{\mathsf{K}}}^{\scriptscriptstyle \mathrm{M}} _j(R)} \xrightarrow{\sim} H^j({\mathbb{Z}}_p(j)(R))
  ,\end{align*}
  where \(\widehat{{\mathsf{K}}}^{\scriptscriptstyle \mathrm{M}}\) is
  the ``improved Milnor \({\mathsf{K}}{\hbox{-}}\)theory'' of
  Gabber-Kerz.
\item
  Comparison to Geisser-Levine: if \(R\) is smooth over a perfect
  characteristic \(p\) field, then
  \begin{align*}
  {\mathbb{Z}}_p(j)(R)/p \xrightarrow{\sim} {\mathbb{R}}\Gamma_\text{ét}(\operatorname{Spec}R; \Omega_{\log}^j)[-j]
  ,\end{align*}
  where \([-j]\) is a right-shift.
\end{enumerate}

\end{theorem}

\begin{remark}

For simplicity, we'll write
\(H^i(j) \coloneqq H^i( {\mathbb{Z}}_p(j)(R))\). The spectral sequence
looks like the following:

It converges to \(K^\text{ét}_{-i-j}(R;{\mathbb{Z}}/p)\). The 0-column
is
\({ \overbrace{ {\mathsf{K}}^{\scriptstyle\mathrm{M}} _{-j}(R)}^{\widehat{\hspace{4em}}} }\),
and we understand the 1-column: we have
\begin{align*}
H^{j+1} \xrightarrow{\sim} \varprojlim_{r} \tilde v_r(j)(R)
.\end{align*}
where \(\tilde v_r(j)(R)\) are the mod \(p^r\) weight \(j\)
Artin-Schreier obstruction. For example,
\begin{align*}
\tilde v_1(j)(R) \coloneqq
\operatorname{coker}\qty{ 
1- C^{-1}: \Omega^j_{R/pR} \to {\Omega^j_{R/pR} \over {{\partial}}\Omega^{j-1}_{R/pR}}
}
= { R \over pR + \left\{{ a^p-a {~\mathrel{\Big|}~}a\in R }\right\} } 
.\end{align*}
These are weird terms that capture some class field theory and are
related to the Tate and Kato conjectures.

\end{remark}

\begin{theorem}[(continued)]

If \(R\) is local, then the 3rd quadrant of the above spectral sequence
gives an Atiyah-Hirzebruch spectral sequence converging to
\({\mathsf{K}}_{-i-j}(R; {\mathbb{Z}}_p)\).

\end{theorem}

\begin{remark}

So we get things describing étale \({\mathsf{K}}{\hbox{-}}\)theory, and
after discarding a little bit we get something describing usual
\({\mathsf{K}}{\hbox{-}}\)theory. Moreover, for any local
\(p{\hbox{-}}\)adically complete ring \(R\), we have broken
\({\mathsf{K}}_*(R; { {\mathbb{Z}}_p })\) into motivic pieces.

\end{remark}

\begin{example}[?]

We same that for number fields, \(\operatorname{cohdim}\leq 2\) yields a
simple spectral sequence relating \(K\) groups to Galois cohomology.
Consider now a truncated polynomial algebra \(A = k[T]/T^r\) for
\(k\in\mathsf{Perf}\mathsf{Field}^{\operatorname{ch}= p}\) and let
\(r\geq 1\). Then by the general bounds given in the theorem,
\(H^i(j) = 0\) unless \(0\leq i \leq 2\), using that \(\Omega\) can be
generated by one element. Slightly more work will show \(H^0, H^2\)
vanish unless \(i=j=0\) (so higher weights vanish), since they're
\(p{\hbox{-}}\)torsionfree and are killed by \(p\).

So the spectral sequence collapses:

\begin{center}
\begin{tikzcd}
    &&&&& {} & {} \\
    \\
    {} &&&&& \textcolor{rgb,255:red,214;green,92;blue,92}{H^0(0)} & {H^1(0)} & 0 & 0 &&& {} \\
    &&&& {H^0(1)} & \textcolor{rgb,255:red,214;green,92;blue,92}{H^1(1)} & {H^2(1)} & 0 & 0 \\
    &&& {H^0(2)} & \textcolor{rgb,255:red,214;green,92;blue,92}{H^1(2)} & {H^2(2)} & {H^3(2)} & 0 & 0 \\
    && \ddots & {H^1(3)} & {H^2(3)} & {H^3(3)} & {H^4(3)} & 0 & 0 \\
    &&& \vdots & \vdots & \vdots & \vdots \\
    &&&&& {} & {}
    \arrow[draw={rgb,255:red,214;green,92;blue,92}, dashed, from=4-5, to=5-7]
    \arrow[draw={rgb,255:red,153;green,153;blue,153}, dotted, no head, from=1-6, to=8-6]
    \arrow[draw={rgb,255:red,153;green,153;blue,153}, dotted, no head, from=3-1, to=3-12]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

So the Atiyah-Hirzebruch spectral sequence collapses to

\begin{align*}
{\mathsf{K}}_n\qty{ { K[T] \over  \left\langle{T^r}\right\rangle }, \left\langle{T}\right\rangle}
=
\begin{cases}
 H^1\qty{{\mathbb{Z}}_p\qty{n+1\over 2}} (R) & n \text{ odd}  
\\
0 & n \text{ even}.
\end{cases}
.\end{align*}

When \(r=2\), one can even valuation these nontrivial terms.

\end{example}

\begin{question}

What is the motivic cohomology for regular schemes not over a field?
We'd like to understand this in general.

\end{question}


\printbibliography[title=Bibliography]


\end{document}
