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\title{
\textbf{
    Aspects of motivic cohomology
  }
    \\ {\normalsize Matthew Morrow, IAS/PCMI GSS 2021} \\
  }







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

\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}
