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

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 \(\operatorname{Spec}k\).

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 \(\operatorname{Sym}^2(V {}^{ \vee })\).

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 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.

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 \(H_{\mathrm{zar}}^*(X; {\mathbb{Z}}/\ell(n)) \xrightarrow{\sim} H_\text{ét}^n(X; {\mathbb{Z}}/\ell(n))\).

There are a number of competing notions for the “dimension” of a field.

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

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

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.

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

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 \(m\).

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.

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:

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.

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.

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 \(b_i, c_i\in k^{\times}\).

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

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.

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 \(I^n(k)\).

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.

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 \(\alpha_E = {\operatorname{pt}}\).

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

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.

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.

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.

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

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.

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*}

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*}

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

This is a “strong approximation” argument.

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 \({\mathcal{U}}\rightrightarrows R\).

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

Link to Diagram

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.

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*} .

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 \(S^{\otimes_R 2}\):

Link to Diagram

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.

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*}

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

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.

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:

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 \(X\in {\mathsf{Aff}}{\mathsf{Sch}}\):

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*}

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*}

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

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.

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

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 \(T\).

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*}

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 \({\mathbb{P}}^1\):

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.

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.

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.

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.

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*}

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.

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*}

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

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.

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*}

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.

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

Link to Diagram

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

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.

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*}

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.

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.

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.

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.

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

Link to Diagram

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.

\({\mathsf{SH}}_{/ {S}}\) 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 \({\mathbb{E}}\to ({\mathbb{E}} {}^{ \vee }) {}^{ \vee }\).

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.

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 \(\left\{{f_1 = f_2 = \cdots f_n = 0}\right\}\).

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*}

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

Link to Diagram

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*}

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.

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 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 \({\mathsf{SH}}(k)\).

\({\mathsf{K}}^{\scriptscriptstyle \mathrm{MW}} _*\) 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*}

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*}

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.

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.

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 \(L_x\).

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*}

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 \(\eta\).

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

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

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

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

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

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

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

Some properties:

• This has a well-behaved cotangent complex \(L_f\).

• For a regular embedding, \(L_i \simeq N_u {}^{ \vee }P[1]\) is conormal bundle.⁸

• \(L_p \simeq\Omega_{P/S} \simeq{\mathsf{T}}_p {}^{ \vee }\).

• \(L_{p_i}\) is determined by \(i^* L_p\to L_{p_i} \to L_i\),

• There is a coherent Serre duality related to \(L_f\).

There is also a good notion of pushforward: let \(p:X\to S\) be proper⁹ and \(\mathrm{lci}\).

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

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

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

• \(H\) is \(\operatorname{GL}{\hbox{-}}\)oriented if \(H_Z^n(X) \xrightarrow{\sim} H_Z^V(X)\) with \(n\mathrel{\vcenter{:}}=\operatorname{rank}V\).

  An example is \(H{\mathbb{Z}}, {\mathsf{K}}\), but non examples are \({H\tilde {\mathbb{Z}}}, {\operatorname{KO}}\).

• \(H\) is \({\operatorname{SL}}{\hbox{-}}\)oriented if

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

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

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

Link to Diagram

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

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

See Déglise-Jin-Khan, Bachmann-Wickelgren.

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

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

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

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

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

• It encodes other invariants.
• It breaks into “simpler” pieces that are motivic in nature.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A review of yesterday:

• \({\mathsf{K}}{\hbox{-}}\)theory can be refined by motivic cohomology, i.e. it breaks into pieces. More precisely we have the Atiyah-Hirzebruch spectral sequence, and even better, the spectrum \({\mathsf{K}}(X)\) has a motivic filtration with graded pieces \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)[2j]\).

• The \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)\) correspond to algebraic cycles and étale cohomology mod \(m\), where \(m\) is prime to \(\operatorname{ch}k\), due to Beilinson-Lichtenbaum and Beilinson-Bloch.

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

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

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

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

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

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

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

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

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

Then \(z^j(X, \bullet)\) defines motivic Borel-Moore homology¹⁰ which refines \({\mathsf{G}}{\hbox{-}}\)theory.

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

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

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

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

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

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

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

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

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

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

Link to Diagram

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

We now focus on

• Étale \({\mathsf{K}}{\hbox{-}}\)theory, \({\mathsf{K}}^\text{ét}\)
• mod \(p\) coefficients, even period
• \(p{\hbox{-}}\)adically complete rings

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

Link to Diagram

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

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

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

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