# 1 Preface

(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.

Also, a disclaimer: most of these notes were live-tex’d, and 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 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.

## 1.2 Definitions

• $$K{\hbox{-}}$$theory
• Milnor $$K{\hbox{-}}$$theory
• Witt ring
• Period and index
• Symbols
• Brauer group
• Bloch’s higher Chow groups
• Mixed characteristic
• Weil Cohomology theory
• Betti cohomology
• Rigid cohomology
• Motivic cohomology
• Étale cohomology
• Galois cohomology
• Sites
• The smooth site
• The étale site
• The Nisnevich site
• Difference between big and small sites
• Cartier divisors
• Henselian schemes
• Dévissage
• Purity theorems
• Fiber functor
• Transfers
• Torsors
• Discretely valued fields
• Cotangent complex
• Symbols
• Cohomological symbol
• Tate symbol
• Galois symbol
• Tate twist

## 1.3 Results/Conjectures

• What is $${\mathsf{K}}_n({\mathbb{Z}})$$?
• Kummer-Vandiver conjecture
• For $$h_k$$ the class number of the maximum real subfield $$K$$ of the $$p$$th cyclotomic fields, $$p\notdivides h_k$$.
• Milnor conjecture (proved)
• Relates $$H_{ \mathsf{Gal}}$$ to to $${\mathsf{K}}^{\scriptstyle\mathrm{M}} /2$$ and quadratic forms.
• Bloch-Kato conjecture, i.e. Norm-Residue isomorphism (proved?)
• 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*}

• Generalized Milnor’s conjecture to odd primes.

• Closely related to Beilinson-Lichtenbaum

• Standard conjectures on algebraic cycles
• Beilinson-Soulé conjecture
• Support in degrees $$d<0$$ for $${\mathbb{Z}}_{ \mathrm{mot}} (j)$$.
• Currently unknown
• Gersten conjecture
• Support in degrees $$d\leq 2j$$ for $${\mathbb{Z}}_{ \mathrm{mot}} (j)$$.
• Beilinson-Lichtenbaum conjecture
• 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*}
• Vanishing and rigidity conjectures
• Kummer–Vandiver conjecture
• Tate conjecture
• Kato conjecture
• Bass’ finite generation conjecture
• Morrow: for $$R$$ a regular ring of finite type over $${\mathbb{Z}}$$, $${\mathsf{K}}_n(R)$$ is finitely generated.
• 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.
• Generalizes Dirichlet’s unit theorem: $${\mathsf{K}}_1(R)$$ is finitely-generated for $${\mathcal{O}}$$ the ring of integers of a number field $$K$$.
• Parshin’s conjecture
• Vanishing of higher $${\mathsf{K}}{\hbox{-}}$$groups for smooth varieties over finite fields.
• Quillen-Lichtenbaum conjecture
• Hilbert 90

## 1.4 Generic Notes

• $${\mathsf{K}}^{\scriptstyle\mathrm{M}}$$ 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.
• Some motivations for $${\mathsf{K}}{\hbox{-}}$$theory:
• Special values of $$L{\hbox{-}}$$functions
• $${\mathsf{K}}_0$$:
• AG: Grothendieck-Riemann-Roch, intersection theory on algebraic varieties.
• NT: $${\operatorname{Pic}}(R)$$, the class group $${ \operatorname{Cl}} (R)$$, class field theory
• Higher regulators
• Embeddings of number fields into $${\mathbb{R}}, {\mathbb{C}}$$.
• Whitehead torsion, used in surgery classification of manifolds and the Poincaré conjecture in $$\dim \geq 5$$.
• Sheaf cohomology of a $${\mathsf{K}}{\hbox{-}}$$theory sheaf recovers Chow (Bloch’s formula)
• The Dennis trace relates $${\mathsf{K}}\to {\operatorname{HH}}$$, and $${\operatorname{THH}}$$ yields an intermediate step.

### 1.4.1 Major Objects

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

$${\mathsf{K}}{\hbox{-}}$$theory:

• $${\mathsf{K}}_*$$: $${\mathsf{K}}{\hbox{-}}$$theory.
• $${\mathsf{K}}_0(F) = {\mathbb{Z}}$$, $${\mathsf{K}}_1(F) = F^{\times}$$.
• $${\mathsf{K}}_0(R) = \mathop{\mathrm{Hom}}_{\mathsf{Top}}(\operatorname{Spec}R, {\mathbb{Z}})$$. If $$R$$ is a domain, $${\mathsf{K}}_0(R) = {\mathbb{Z}}$$.
• $${\mathsf{K}}_0({\mathcal{O}}) \in \operatorname{Ext} ({\mathbb{Z}}, { \operatorname{Cl}} ({\mathcal{O}}))$$, $${\mathsf{K}}_1({\mathcal{O}}) = {\mathcal{O}}^{\times}$$.
• $${\mathsf{K}}_1(R) = \operatorname{GL}(R)$$.
• 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*}
• $${\operatorname{KO}}_*$$: Hermitian $${\mathsf{K}}{\hbox{-}}$$theory.

# 2 Danny Krashen, Talk 1 (Monday, July 12)

## 2.1 Intro

Abstract:

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.

References:

Fix a field $$k_0 \in \mathsf{Field}$$, we’ll consider extensions $$k\in\mathsf{Field}_{/ {k_0}}$$.

## 2.2 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; {-}) \mathrel{\vcenter{:}}={\mathbb{R}}^*({-})^{G_k} .\end{align*}

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 $$\mu_\ell^{\otimes 0} \mathrel{\vcenter{:}}={\mathbb{Z}}/n$$. It is known that $$H_{ \mathsf{Gal}} ^*(G_k; \mu^{\otimes 0}) = {\mathbb{Z}}/n$$.

We’ll define symbols \begin{align*} (a_1, \cdots, a_n) \mathrel{\vcenter{:}}=(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 $$({ {a}_1, {a}_2, \cdots, {a}_{n}})_\ell$$.

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 $$\operatorname{Spec}k$$.

## 2.3 Milnor K-Theory

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

• $${\mathsf{K}}^{\scriptstyle\mathrm{M}} _0(k) = {\mathbb{Z}}$$
• $${\mathsf{K}}^{\scriptstyle\mathrm{M}} _1(k) = k^m$$, written additively as elements $$\left\{{a}\right\}$$ on the left-hand side, so $$\left\{{a}\right\} + \left\{{b}\right\} \mathrel{\vcenter{:}}=\left\{{ab}\right\}$$.
• 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*}
• 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*}
• 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 Norm-Residue isomorphism (formerly the 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*}

## 2.4 Witt Ring

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) \mathrel{\vcenter{:}}= b(x,x) &\mapsfrom b(x, y) \\ q &\mapsto b_q(x,y) \mathrel{\vcenter{:}}={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 nondegenerate if there is an induced isomorphism: \begin{align*} V &\xrightarrow{\sim} V {}^{ \vee }\\ v &\mapsto b_q(v, {-}) .\end{align*}

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

Let $${\mathsf{QuadForm}}_{/ {k}}$$ be the category of 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 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 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).

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.

The fundamental ideal $$I(k) {~\trianglelefteq~}W(k)$$ is the ideal of even dimensional forms, and set $$I^n(k) \mathrel{\vcenter{:}}=(I(k))^n$$. 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 $$n{\hbox{-}}$$fold Pfister forms \begin{align*} \left\langle{\left\langle{ a }\right\rangle}\right\rangle &\mathrel{\vcenter{:}}=\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 &\mathrel{\vcenter{:}}=\prod_{i=1}^n \left\langle{\left\langle{ a_i }\right\rangle}\right\rangle .\end{align*}

The 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.

## 2.5 Motivic Cohomology

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 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.

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\mathrel{\vcenter{:}}=\operatorname{Spec}k .\end{align*} These relate to Galois cohomology in the following ways:

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

## 2.6 Dimension

There are a number of competing notions for the “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*}

We define its 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) \mathrel{\vcenter{:}}=\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 $$m>n$$.

$$\operatorname{cohdim}(k) = \dim(k)$$ 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, $$\operatorname{cohdim}(k) = \infty$$.

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) \mathrel{\vcenter{:}}=\min\left\{{n{~\mathrel{\Big|}~}k \text{ is } C_n}\right\} .\end{align*}

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*}

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 $$T_n{\hbox{-}}$$rank is defined as \begin{align*} T_n{\hbox{-}}\operatorname{rank}(k) \mathrel{\vcenter{:}}=\min\left\{{n {~\mathrel{\Big|}~}k \text{ is } T_n}\right\} .\end{align*}

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.

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

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?

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 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.

## 2.7 Structural Problems in Galois Cohomology

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

### 2.7.1 Period-Index Problems

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

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

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 $$m$$.

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 $$\mathop{\mathrm{ind}}\alpha \divides \ell^m$$?

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*}

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 $$\mathop{\mathrm{per}}k = \mathop{\mathrm{ind}}k$$.

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.

### 2.7.2 Symbol Length Problem

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:

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 $$\alpha$$?

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 $$n\geq 4$$.

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.

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.

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.

# 3 Danny Krashen, Talk 2 (Tuesday, July 13)

## 3.1 Setup

Fix a $$k_0\in\mathsf{Field}$$.

Outline

• Arithmetic problems: consider “complexity” of cohomology or algebraic structures (Witt group, symbol length, index of classes).

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

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

## 3.2 Symbol Length

For a particularly concrete problem, consider \begin{align*} \alpha\in H^2(k; \mu_\ell) \subseteq H^2(k; {\mathbb{G}}_m)[\ell] \mathrel{\vcenter{:}}=\mathop{\mathrm{Br}}(k)[\ell] ,\end{align*} i.e. this is a subgroup of the $$\ell{\hbox{-}}$$torsion of the Brauer group. Suppose we know \begin{align*} \mathop{\mathrm{ind}}\alpha \mathrel{\vcenter{:}}=\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 $$b_i, c_i\in k^{\times}$$.

How big does $$n$$ have to be?

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:

• $$\ell = n = 2, 3$$: $$r\leq 1$$, so only one symbol is needed.
• $$\ell = n = 4$$: probably $$r\leq 4$$.
• $$\ell = 2, n=4$$: $$r\leq 2$$, a classical results on central simple algebras.
• $$\ell = 2, n=8: r\leq 4$$

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 $$\alpha \in \operatorname{im}\qty{H^2(L;\mu) \to H^2(k; \mu)}$$.

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

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 very low: it is hard to show a symbol can not be written with very few symbols.

## 3.3 Pfister Forms

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 \mathrel{\vcenter{:}}=\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 $$I^n(k)$$.

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?

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

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?

Consider the following objects:

• $$H^2(k; \mu)$$
• $$\mathop{\mathrm{Br}}(k)$$
• $$W(k)$$
• $$I^n(k)$$
• $$q\in I^n(k)$$ with $$\dim q = d$$

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

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*}

Given a functor $$F: {\mathsf{Alg}}_{_{/ {k_0}} }\to {\mathsf{Set}}$$, we say $$\alpha \in F(R)$$ is 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 $$F(R)\to F(k)$$.

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.

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 $$L \mathrel{\vcenter{:}}= k_0({ {a}_1, {a}_2, \cdots, {a}_{n}})$$. 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 $$e_n(q) = 0$$.

## 3.4 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) \mathrel{\vcenter{:}}= \begin{cases} \emptyset & \alpha_L \neq {\operatorname{pt}} \\ \left\{{{\operatorname{pt}}}\right\} & \alpha_L = {\operatorname{pt}}, \end{cases} \end{align*} and define the canonical dimension \begin{align*} \operatorname{candim}(\alpha) = \operatorname{essdim}(\check{F}(\alpha)) .\end{align*}

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 $$\alpha_E = {\operatorname{pt}}$$.

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

So this encodes triviality of $$\alpha$$ into polynomial equations.

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

## 3.5 Splitting Schemes

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

We do know this in special cases:

• $$i=1$$: Yes, these are etale algebras, so finite schemes over $$k$$.
• $$i=2$$: Yes, Danny shows these exist for all twists.
• $$j=1$$: Classical, these are Severi-Brauer varieties.
• For symbols, $$i=3,j=2,\ell$$ a prime: see Merkurjev-Suslin
• For symbols, $$i=4, j=3, \ell =3$$: see Albert algebras
• For symbols, $$\ell$$ prime: this can be done up to prime-to-$$\ell$$ extensions, see Rost’s “Norm Varieties”. Related to Bloch-Kato conjecture.
• For symbols, $$\ell=2$$: see Pfister quadrics.

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.

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.

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.

# 4 Phillippe Gille, Talk 1 (Monday, July 12)

## 4.1 Intro

Abstract:

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.

References:

Notation:

• $${\mathsf{Bun}}_{G}$$ is the category of $$G{\hbox{-}}$$bundles
• $${\mathsf{Bun}}_{\operatorname{GL}_r({\mathbb{R}})}$$ is the category of real vector bundles of rank $$r$$.
• $${ {\mathsf{Bun}}_{\operatorname{GL}_r} }$$ is the category of vector bundles of rank $$r$$ over an unspecified field.
• $$\mathsf{C}_{/ {R}}$$ is an overcategory/slice category of objects over a fixed object $$R$$.
• $${\mathsf{Sch}}_{/ {S}}$$ is the category of schemes over a fixed scheme $$S$$.
• $${\mathsf{Aff}}{\mathsf{Sch}}$$ is the category of affine schemes.
• $${\mathsf{Sch}}_{/ {R}}$$ are the schemes over $$\operatorname{Spec}R$$.
• $${\mathsf{sm}}$$ denotes subcategories of smooth objects
• $$R \left[ { \scriptstyle {\mathfrak{p}}^{-1}} \right]$$ is the localization of $$R$$ at $${\mathfrak{p}}$$.
• $${\mathsf{cAlg}}$$ denotes commutative algebras.
• 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.
• Todo: so any presheaf on $$R{\hbox{-}}$$algebras..?

## 4.2 Serre-Swan and Vector Group Schemes

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

• Vector bundles
• Principal $$G{\hbox{-}}$$bundles
• Principal homogeneous spaces

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.

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.

Let $$R \in \mathsf{CRing}$$ be unital and $$M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}$$. Let $$V(M)$$ denote the affine $$R{\hbox{-}}$$scheme $$V(M) \mathrel{\vcenter{:}}=\operatorname{Spec}\qty{\operatorname{Sym}^\bullet M}$$, 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 vector group scheme of $$M$$.

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

$$V$$ 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*}

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 $$W(M) \mathrel{\vcenter{:}}= V(M {}^{ \vee })$$. Note that finite locally free is a necessary condition.

## 4.3 Vector Bundles over Affine Schemes

A 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.

$$M\to V(M)$$ 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*}

Given a smooth1 map of affine schemes \begin{align*} (X\to Y) \mathrel{\vcenter{:}}=(\operatorname{Spec}S\to \operatorname{Spec}R) && r\mathrel{\vcenter{:}}=\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*}

Consider the real sphere \begin{align*} Z \mathrel{\vcenter{:}}=\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 $${\mathbb{R}}$$.

## 4.4 Linear Groups

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) \mathrel{\vcenter{:}}=\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 $$M\mathrel{\vcenter{:}}= M^r$$ recovers the usual $$\operatorname{GL}_n(R)$$ for $$n\mathrel{\vcenter{:}}=\operatorname{rank}_R M$$. Note that local freeness is necessary for representability by a group scheme here.

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*}

Take an affine cover \begin{align*} {\mathcal{U}}\mathrel{\vcenter{:}}=\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) \mathrel{\vcenter{:}}=\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 $$\operatorname{Spec}R$$.

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*}

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

## 4.5 Classification of Modules over a Dedekind Ring

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.

Write $$n= \prod r_i$$ to get a similar map, sometimes called the 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*}

We set $$\operatorname{det}(V) \mathrel{\vcenter{:}}=\operatorname{det}_*(V) \mathrel{\vcenter{:}}=\bigwedge\nolimits^r V$$, the determinant bundle.

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

Let $$R$$ be a Dedekind ring2 , 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.

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

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 \mathrel{\vcenter{:}}=\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 $$\widehat{K} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right] \mathrel{\vcenter{:}}= K\otimes_R \widehat{R} \left[ { \scriptstyle {\mathfrak{p}}_i^{-1}} \right]$$ 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) \mathrel{\vcenter{:}}= \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*}

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.

We can assume $$\operatorname{det}V(M)$$ is trivial to get \begin{align*} g_i \in \ker \operatorname{det}_* \mathrel{\vcenter{:}}=\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.

This is a “strong approximation” argument.

## 4.6 Replacing the Zariski Topology

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

• $$q(\lambda x) = \lambda^2 q(x)$$ for all $$\lambda\in R, x\in M$$,
• 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*}

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

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

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) \mathrel{\vcenter{:}}=\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 $${\mathcal{U}}\rightrightarrows R$$.

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

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

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) \mathrel{\vcenter{:}}=\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

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.

A 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*}

Setting $$S\mathrel{\vcenter{:}}={\textstyle\coprod}S_i$$, this says $$S$$ is a faithfully flat finitely presented $$R{\hbox{-}}$$algebra. Note that the Zariski cover is a flat cover.

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)\mathrel{\vcenter{:}}=\colim H^1(S_{/ {R}} ; G) .\end{align*} .

A 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

• 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 $$T\in \mathsf{Ring}_{/ {R}}$$.

• 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.

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

Setting $$X\mathrel{\vcenter{:}}= G$$ with $$G\curvearrowright G$$ by right-translation yields the split 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 trivial torsor.

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

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 first fppf cohomology to be isomorphism classes of $$G{\hbox{-}}$$torsors: \begin{align*} H^1_\mathrm{\operatorname{fppf}}(R; G) \mathrel{\vcenter{:}}={\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*}

There is a 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*}

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 $$S^{\otimes_R 2}$$:

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.

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 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.

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*}

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

A 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:

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

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 canonical descent datum \begin{align*} {\mathrm{can}}_M: p_1^*(M\otimes_R T) \xrightarrow{\sim} p_2^*(M\otimes_R T) .\end{align*}

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*}

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

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*}

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

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.

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

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:

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 twists $${ {}^{Y} E }$$ and $${ {}^{G} E }$$. In general, we can twist $$G{\hbox{-}}$$schemes equipped with an amply invertible $$G{\hbox{-}}$$linearized line bundle.

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

# 5 Phillippe Gille, Talk 2 (Tuesday, July 13)

## 5.1 Intro

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

• $$V(M) \mathrel{\vcenter{:}}=\operatorname{Spec}\operatorname{Sym}^\bullet M$$ which represents \begin{align*} S \mapsto \hom_{{\mathsf{(S)}{\hbox{-}}\mathsf{Mod}}}(M\otimes_R S, S) ,\end{align*} is the vector group scheme of $$M$$.
• $$W(M)\mathrel{\vcenter{:}}= V(M {}^{ \vee })$$ which represents $$M\otimes_R {-}$$ (and doesn’t seem to have a name).
• 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 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*}
• For any $$G{\hbox{-}}$$torsor $$E$$ and any $$Y\in G{\hbox{-}}{\mathsf{Sch}}$$ with an ample invertible $$G{\hbox{-}}$$linearized bundle3 , one can similarly define twists $${ {}^{E} Y }$$.
• For $$T$$ a faithfully flat extension of $$R$$, the Amitsur resolution is given by \begin{align*} M\to M \otimes_R {\mathbb{T}}(T)^\bullet && \text{where } {\mathbb{T}}(V)^\bullet \mathrel{\vcenter{:}}= V \oplus V^{\otimes 2} \oplus \cdots .\end{align*} I.e., this resolves $$M$$ by the tensor algebra (or free algebra) on $$T$$.

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

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:

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

## 5.2 Galois Covers

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 Galois cover.

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

Another nice case is when $$\Gamma$$ is the automorphism group associated to some algebraic structure, i.e. when one has forms. For example, take $$\Gamma \mathrel{\vcenter{:}}={\operatorname{O}}_{2n} = \mathop{\mathrm{Aut}}(q_{\operatorname{hyp}})$$ 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.

Take $$\Gamma \mathrel{\vcenter{:}}= S_n$$ 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.

## 5.3 Flat Quotients

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

• 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
• 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 couvrant en français).
• $${\mathbb{G}}_m = \operatorname{GL}_r /_{{\operatorname{Flat}}}\, {\operatorname{SL}}_r$$.
• $${\mathbb{G}}_m = {\mathbb{G}}_m /_{{\operatorname{Flat}}}\, \mu_d$$.

Let $$X$$ be the flat quotient of $$H$$ by $$G$$.

1. $$H\to X$$ is a $$G{\hbox{-}}$$torsor
2. 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 characteristic map, which arises naturally from base change.

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*}

$${\operatorname{SL}}_n{\hbox{-}}$$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 $$\operatorname{det}(M)$$.

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.

An 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.

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

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*}

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

## 5.4 Galois Cohomology

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 semilinear means that \begin{align*} \varphi(\sigma) (\lambda \cdot n) = \sigma(\lambda) \cdot \rho( \sigma)(n) .\end{align*}

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

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

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 loop torsors, which are closely related to representation theory in $${\mathsf{Alg}}{\mathsf{Grp}}$$.

The main topic is affine curves, and these are special cases of Dedekind rings. Let $$R$$ be Dedekind with $$K\mathrel{\vcenter{:}}=\operatorname{ff}(R)$$ 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.

If $$G$$ is a semisimple simply connected4 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*}

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 $$H^1_{\mathrm{zar}}(R; E_8) = 1$$.

## 5.5 Curves Over an Algebraically Closed Field

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 $$H^1_\mathrm{\operatorname{fppf}}(C; G) = 1$$.

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.

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 $$T\mathrel{\vcenter{:}}=\tilde T/\ker f$$ is a maximal torus in $$G$$, so let $$B \subseteq G$$ be a Borel containing $$T$$.

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

Letting $$E$$ be a $$C{\hbox{-}}$$torsor under $$G$$, then the idea is to introduce the twisted $$C{\hbox{-}}$$scheme $$Y\mathrel{\vcenter{:}}= E(G/B)$$, 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*}

## 5.6 Affine Line

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 constant.

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 $${\mathbb{P}}^1$$:

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.

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 $$F \mathrel{\vcenter{:}}= k(u)$$.

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.

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.

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*}

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 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.

# 6 Frédéric Déglise, Talk 1 (Wednesday, July 14)

## 6.1 Intro

References:

Abstract:

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.

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 weights in relation to the Weil conjectures, and served as an impetus in the early 70s for 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.

An outline for today:

1. Sheaves with transfers, which are modeled on étale homotopy sheaves

2. Homotopy sheaves over perfect fields

3. Motivic complexes

There are three main notions for étale sheaves:

1. Sheaves with transfers (see algebraic cycles),
2. The (big) smooth Nisnevich site,
3. $${\mathbb{A}}^1{\hbox{-}}$$homotopy invariance.

## 6.2 Setting up sheaves with transfers

We’ll fix $$S$$ a regular Noetherian scheme.

### 6.2.1 Finite Correspondences

For $$X,Y\in {\mathsf{sm}}{\mathsf{Sch}}_{/ {S}}$$, a 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 category of finite correspondences over $$S$$: \begin{align*} \mathsf{C} &\mathrel{\vcenter{:}}={\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} \\ \mathrm{Ob}(\mathsf{C}) &\mathrel{\vcenter{:}}={\operatorname{Ob}}({\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} ) \\ \mathsf{C}(X, Y) &\mathrel{\vcenter{:}}= 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.

Writing $$XYZ \mathrel{\vcenter{:}}= X { \underset{\scriptscriptstyle {S} }{\times} } Y { \underset{\scriptscriptstyle {S} }{\times} } Z$$, 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.

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

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

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*}

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 transpose of $$f$$.

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

• 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 pullback for finite correspondences.
• 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 direct images for finite correspondences.

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) \mathrel{\vcenter{:}}={\mathbb{Z}}^{\mathrm{tr}}_{/ {S}} (X) \end{align*} landing in a cocomplete abelian category extending the 6 functors.

### 6.2.2 Presheaves with transfers

A 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 category of presheaves with transfers over $$S$$: \begin{align*} \mathsf{C} &\mathrel{\vcenter{:}}={\mathrm{tr}} \underset{ \mathsf{pre} } {\mathsf{Sh} }_{/ {S}} \\ {\operatorname{Ob}}(\mathsf{C}) &\mathrel{\vcenter{:}}=\text{Presheaves with transfers } {\mathcal{F}}\\ \mathsf{C}({\mathcal{F}}, {\mathcal{G}}) &\mathrel{\vcenter{:}}=\text{Natural transformations } \eta: {\mathcal{F}}\to{\mathcal{G}} .\end{align*}

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 transfer is the additional following operation on $${\mathcal{F}}$$, yielding a “wrong way” map: \begin{align*} f_* \mathrel{\vcenter{:}}={\mathcal{F}}(f^t) \in {\mathsf{Ab}}{\mathsf{Grp}}( {\mathcal{F}}(Y), {\mathcal{F}}(X)) .\end{align*}

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

### 6.2.3 Nisnevich Sheaves

A 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 $$\kappa(w) / \kappa(x)$$.

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:

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

### 6.2.4 Sheaves 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 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 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 $${ \left.{{a^{\mathrm{tr}}({\mathcal{F}})}} \right|_{{{\mathsf{sm}}{\mathsf{Sch}}_{/ {S}} }} } = \left( F\circ \gamma \right)^{\scriptscriptstyle \mathrm{sh}}$$.

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 $${\mathsf{Sh}}({\mathsf{Sch}}^{\mathrm{Nis}}_{/ {S}} )$$.

• For $$X_{/ {S}}$$ smooth: $${\mathbb{Z}}_s^{\mathrm{tr}}(X) \mathrel{\vcenter{:}}= c_S({-}, X)$$.
• $${\mathbb{G}}_m({-}): X_{/ {S}} \to {\mathcal{O}}_X(X)^{\times}$$.
• For $$A\in {\mathsf{Ab}}{\mathsf{Var}}_{/ {k}}$$, $$\mathop{\mathrm{Hom}}({-}, A)$$.

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

• $$H^n({-})$$
• $${\mathsf{K}}{\hbox{-}}$$theory

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

• $$f_*(F) \mathrel{\vcenter{:}}= F \circ f^*$$, which yields a base change/direct image adjunction: \begin{align*} \adjunction{f^*}{f_*}{{\mathrm{tr}}{\mathsf{Sh}}_{/ {S}} }{{\mathrm{tr}}{\mathsf{Sh}}_{/ {T}} } .\end{align*}

• $$p:T\to S$$ yields a 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 exceptional direct image

• $${}_{h}\otimes$$ on $${\mathrm{tr}}{\mathsf{Sh}}_{/ {S}}$$ yields a 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.

## 6.3 Homotopy and Cohomology

### 6.3.1$${\mathbb{A}}^1{\hbox{-}}$$invariance and Homotopic Morphisms

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 $${\mathbb{A}}^1{\hbox{-}}$$invariant or a 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 $${\mathsf{HI}}^{\mathrm{tr}}(S)$$.

Let $$\alpha, \beta \in {\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (X, Y)$$ be two morphisms. We then say $$\alpha$$ 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) \mathrel{\vcenter{:}}={\mathsf{Cor}}{\mathsf{Sch}}_{/ {S}} (X, Y)/\sim .\end{align*}

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

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

• $$\tilde C_{/ {S}}$$ is proper and normal,
• $$C\subseteq \tilde C$$ is open/dense,
• $$C_\infty \mathrel{\vcenter{:}}=\tilde C\setminus C$$ admits an affine open neighborhood.

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 $$X { \underset{\scriptscriptstyle {S} }{\times} } \tilde C$$.

### 6.3.2 Cohomology of Perfect Fields

Fix $$k \in \mathsf{Perf}\mathsf{Field}$$, then a function field $$E$$ over $$k$$ is a separable finitely generated field extension $$E_{/ {k}}$$. One can define the 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}} ) \mathrel{\vcenter{:}}=\colim_{A_{/ {k}} \leq E_{/ {k}} } F(\operatorname{Spec}A) .\end{align*} This yields a fiber functor: it is exact and commutes with coproducts.

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*}

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*}

### 6.3.3 Homotopy Invariance

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.

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.

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*}

### 6.3.4 Relation to Chow

Write $${\mathbb{S}}^{n} \mathrel{\vcenter{:}}={\mathbb{G}}_m{ {}^{ \scriptstyle {}_{h} {\otimes_{}^{n}} } }$$, 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.

# 7 Frédéric Déglise, Talk 2 (Thursday, July 15)

## 7.1 Intro

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.

## 7.2 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) \mathrel{\vcenter{:}}={\operatorname{hofib}}( F(X) \to F(X\setminus Z)) .\end{align*} Then the (pointed) $${\mathbb{A}}^1$$ homotopy category of $$S$$, denoted $${\mathsf{ho}_*^{\scriptstyle {\mathbb{A}}^1}}_{/ {S}}$$, consists of such functors $$F$$ that satisfy

• 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*}

• 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*}

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

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

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

We can use infinity categorical localization theory to build this category. For a scheme $$S$$, view a 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] \mathrel{\vcenter{:}}=\mathsf{C} \left[ { \scriptstyle W^{-1}} \right] \\ \\ W &\mathrel{\vcenter{:}}=\left\{{ {\mathbb{Z}}_S^*({\mathbb{A}}^1_{/ {X}} ) \to {\mathbb{Z}}_S^*(X) {~\mathrel{\Big|}~}X\in {\operatorname{Ob}}(\mathsf{C}) }\right\} .\end{align*}

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 $${\mathbb{Z}}_S^{\mathrm{tr}}({\mathbb{A}}^1_{/ {X}} ) \to {\mathbb{Z}}_S^{{\mathrm{tr}}}(X)$$.

## 7.3$${\mathbb{A}}^1{\hbox{-}}$$locality

$${\mathbb{A}}^1{\hbox{-}}$$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*}

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) \mathrel{\vcenter{:}}= \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.

$$K$$ 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 $$n$$

Define standard cosimplicial scheme as \begin{align*} \Delta^n\mathrel{\vcenter{:}}=\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 Suslin singular complex is the complex of sheaves defined as \begin{align*} \underline{C_{*}^S}(K), && {\mathsf{\Gamma}\qty{X_{/ {S}} } } \mathrel{\vcenter{:}}={ \operatorname{Tot} }^{\Pi}K(\Delta^\bullet { \underset{\scriptscriptstyle {k} }{\times} } X) .\end{align*} for $$X\in{ {\mathsf{sm}}{\mathsf{Sch}}}_{/ {S}}$$.

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

## 7.4 Motives

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

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

$${\mathbb{Z}}(1) = {\mathbb{G}}_m[-1]\in [0, 1]$$ 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 $$n$$.

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.

## 7.5 Motivic Cohomology

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

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*}

## 7.6 Stable Six Functors

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*}

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

Let $$i: Z\hookrightarrow S$$ be closed and $$U\mathrel{\vcenter{:}}= S\setminus Z$$ 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.

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.

## 7.7 Stabilization

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}({-}) \mathrel{\vcenter{:}}={\mathbf{R}}\underline{\mathop{\mathrm{Hom}}}({\mathbb{P}}^1, {-}) ,\end{align*} where one needs to derive this construction.

The 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.

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 $${\mathbb{S}}^{n, m} \mathrel{\vcenter{:}}= S^n \wedge{\mathbb{G}}_m^m$$, we have \begin{align*} {\mathbb{P}}^1 &\simeq{\mathbb{S}}^{1, 1} && \mathrel{\vcenter{:}}= S^1 \wedge{\mathbb{G}}_m \\ {\mathbb{A}}^n\setminus\left\{{0}\right\}&\simeq{\mathbb{S}}^{n-1, n} && \mathrel{\vcenter{:}}= S^n \wedge{\mathbb{G}}_m{ {}^{ \scriptstyle\otimes_{k}^{n} } } .\end{align*}

A concrete model is given by sequences of objects, called $${\mathbb{P}}^1{\hbox{-}}$$spectra. These are sequences of pointed spaces $$X\mathrel{\vcenter{:}}=(X_m)$$ 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.

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

1. $$f_!$$ is compatible with composition.
2. If $$f$$ is proper then there is am isomorphism $$\eta: f_! \xrightarrow{\sim} f_*$$.
3. 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) \mathrel{\vcenter{:}}={\Sigma}^\infty(T_f/T_f {}^{ \vee }) \end{align*} is its associated Thom space.

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

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

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.

## 7.8 Rational Homotopy

$${\mathsf{SH}}_{/ {S}}$$ 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*}

• The plus part is characterized by the algebraic Hopf map $$\eta$$ acting by zero, $${\varepsilon}= -1$$
• The minus by $$\eta$$ being invertible and $${\varepsilon}= +1$$

For $$S$$ regular, the plus part is equivalent to $$\qty{ {\mathsf{DM}}_{/ {S}} }_{\mathbb{Q}}$$. Writing $$S^0 \mathrel{\vcenter{:}}= S\otimes_{\mathbb{Z}}{\mathbb{Q}}$$, 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 $$\tilde {\mathbb{S}}^{n, i} \mathrel{\vcenter{:}}= S^{n-i} \wedge{\mathbb{G}}_m{ {}^{ \scriptstyle\otimes_{k}^{i} } }$$, one can define cohomotopy groups \begin{align*} \qty{\pi^{n, i}_{/ {S}} }_{\mathbb{Q}} &\mathrel{\vcenter{:}}= [ {\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*}

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 $${\mathbb{E}}\to ({\mathbb{E}} {}^{ \vee }) {}^{ \vee }$$.

# 8 Kirsten Wickelgren, Talk 1 (Wednesday, July 14)

## 8.1 Intro

Abstract:

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?”

References/Background:

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.

## 8.2 Classical Theory

First some classical homotopy theory. The sphere can be defined as \begin{align*} S^n \mathrel{\vcenter{:}}=\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 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 $$\deg_{q_i} f \mathrel{\vcenter{:}}=\deg \mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu$$.

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 $$J_f \mathrel{\vcenter{:}}=\operatorname{det}\qty{{\partial}f_i \over {\partial}x_i}$$. 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 $$\left\{{f_1 = f_2 = \cdots f_n = 0}\right\}$$.

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*}

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

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*}

Let $$X \mathrel{\vcenter{:}}={\operatorname{Gr}}(1, 3)_{/{\mathbb{C}}}$$, 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:

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

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 $$e({\mathcal{E}}) = 2$$.

## 8.3 Over arbitrary fields: Grothendieck-Witt

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 } \mathrel{\vcenter{:}}={{\operatorname{GW}}(k) \over {\mathbb{Z}} { \left[ {\left\langle{1}\right\rangle + \left\langle{-1}\right\rangle} \right] } } .\end{align*} There is a 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

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

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

The 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.

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*}

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.

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 \mathrel{\vcenter{:}}=\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, Brazelton-McKean-Pauli give formulas in terms of Bézoutians.

## 8.4 Homotopy

Recall that \begin{align*} X\wedge Y \mathrel{\vcenter{:}}={ 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 \mathrel{\vcenter{:}}= S^1 \wedge X .\end{align*} In $${\mathbb{A}}^1$$ homotopy theory we declare $${\mathbb{A}}^1 \simeq{\operatorname{pt}}$$.

We can take a pushout of the following form:

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 \mathrel{\vcenter{:}}= S^1\wedge{\mathbb{G}}_m \simeq{\mathbb{P}}^1 .\end{align*}

We have \begin{align*} {\mathbb{G}}_m \mathrel{\vcenter{:}}=\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*}

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*}

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

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 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*}

$${\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*$$ 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

• $$[u][1-u] = 0$$, the Steinberg relations,
• $$[ab] = [a] + [b] + \eta[a][b]$$,
• $$[a]\eta = \eta[a]$$,
• $$\eta q_{\operatorname{hyp}}= 0$$ for $$q_{\operatorname{hyp}}\mathrel{\vcenter{:}}=\eta[-1] + 2$$

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}}\mathrel{\vcenter{:}}=\left\langle{1}\right\rangle + \left\langle{-1}\right\rangle &\rightleftharpoons 1 + 1 + \eta[-1] .\end{align*}

$$[a]$$ 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.

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*}

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

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*}

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*}

## 8.5 Big Problems

Notation: we’ll write \begin{align*} \bigoplus_{n\in {\mathbb{Z}}} \pi_{n, n} {\mathbb{S}}\mathrel{\vcenter{:}}=\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. 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.

What is $$[{\mathbb{P}}^n/{\mathbb{P}}^{n-1}, {\mathbb{P}}^n/{\mathbb{P}}^{n-1}]$$ for more general rings? 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*}

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

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

## 8.6 Counting Things

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.

A bundle $$V\to X$$ is 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 relatively oriented when $$\mathop{\mathrm{Hom}}(\operatorname{det}TX, \operatorname{det}V)$$ is oriented, where $$\operatorname{det}({-}) = \bigwedge\nolimits^{\text{top}}({-})$$.

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.

## 8.7 Euler Numbers

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

Suppose

• $$V$$ 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
• For all $$x\in \left\{{f = 0}\right\}$$, the composite map \begin{align*} Tf \mathrel{\vcenter{:}}=\qty{T_x X \to T_{f(x)} V \xrightarrow{\sim} T_x X \oplus V_x \xrightarrow{p_2} V_x} && f(x) \mathrel{\vcenter{:}}=(x, 0) \end{align*} has nonvanishing determinant.

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

• Choosing local Nisnevich coordinates on $$X$$ ,
• Choosing local trivializations of $$V$$ which are “compatible” with $$\phi$$,

Then locally writing \begin{align*} f: {\mathbb{A}}^d \to {\mathbb{A}}^d \implies J_f \mathrel{\vcenter{:}}=\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 \mathrel{\vcenter{:}}=\operatorname{Tr}_{\kappa(x) _{/ {k}} } \left\langle{J_f(x)}\right\rangle .\end{align*}

Equivalently, $$T_x f\in \mathop{\mathrm{Hom}}(T_x X, V_x)$$ and we can define \begin{align*} J_f(x) \mathrel{\vcenter{:}}=\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 $$L_x$$.

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.

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?

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 $$n(V, f) = \pi_* e(V, f)$$.

# 9 Kirsten Wickelgren, Talk 2 (Friday, July 16)

## 9.1 Intro

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 $$J_f \mathrel{\vcenter{:}}=\operatorname{det}{\frac{\partial f_i}{\partial x_j}\,}$$ and write \begin{align*} \deg_x f = \begin{cases} +1 & J(q_i) > 0 \\ -1 & J(q_i) < 0. \end{cases} \end{align*}

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

Over $$k\mathrel{\vcenter{:}}={\mathbb{R}}$$, 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 \mathrel{\vcenter{:}}={{\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 Gorenstein5, there is an isomorphism $$\mathop{\mathrm{Hom}}_k(Q, k) \xrightarrow{\sim} Q$$, which we can take to be the bilinear form.6

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}}} &\mathrel{\vcenter{:}}=\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 $$\eta$$.

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 $$h \mathrel{\vcenter{:}}={ \begin{bmatrix} {1} & {0} \\ {0} & {-1} \end{bmatrix} }$$.

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

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

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

## 9.2$${\mathbb{A}}^1{\hbox{-}}$$Milnor numbers

For $$\operatorname{ch}k\neq 2$$, the simplest type of singularity is a node, defined over $$\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$$ as a point $$p\in X$$ such that7 \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 hypersurface singularity is a point $$p\in \left\{{f=0}\right\} \subseteq X$$.

Let $$k\mathrel{\vcenter{:}}={\mathbb{C}}$$. If you vary $$X$$ in a family \begin{align*} X_+ \mathrel{\vcenter{:}}=\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 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 \mathrel{\vcenter{:}}=\deg_p^{{\mathsf{Top}}}\operatorname{grad}f .\end{align*}

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 $$a\in L^{\times}/ (L^{\times}){ {}^{ \scriptscriptstyle\times^{2} } }$$.

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

• $$x_1^2 +x_2^2 = 0$$, yielding a non-split node and non-rational tangent directions

• $$x_1^2-x_2^2=0$$, yielding a split node and rational tangent directions

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 \mathrel{\vcenter{:}}=\kappa(p) .\end{align*} The type $$p$$ is defined as \begin{align*} {\operatorname{type}}(p) \mathrel{\vcenter{:}}=\operatorname{Tr}_{L_{/ {k}} } \left\langle{2^n \prod_{i=1}^n a_i}\right\rangle \in {\operatorname{GW}}(k) .\end{align*}

The $${\mathbb{A}}^1{\hbox{-}}$$Milnor number is defined as \begin{align*} M_p \mathrel{\vcenter{:}}=\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 $$\left\{{{ {a}_1, {a}_2, \cdots, {a}_{n}} }\right\}$$.

Note: the second equality is due to Kass-Wickelgren.

Let $$f(x,y)\mathrel{\vcenter{:}}= x^3-y^2$$ with $$\operatorname{ch}k\neq 2,3$$, then

• $$p=(0, 0) \in \left\{{f=0}\right\}$$,
• $$\nabla f = (3x^2, -2y)$$, 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.

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