# Phillippe Gille, Talk 2 (Tuesday, July 13) ## Intro :::{.remark} Let $M\in \rmod$. Reminders of notation - $V(M) \da \spec \Sym^\bullet M$ which represents \[ S \mapsto \hom_{\mods{(S)}}(M\tensor_R S, S) ,\] is the **vector group scheme** of $M$. - $W(M)\da V(M\dual)$ which represents $M\tensor_R \wait$ (and doesn't seem to have a name). - For $Y\in G\dash\Aff\Sch_{/R}$ (affine schemes with a left $G\dash$action), $Y_g\in \Aff\Sch_{/R}$ is the **twist** of $Y$ by the 1-cocycle $g$ defined by the action map: \[ g: Y \fiberproduct{R} S^{\tensor_R 2} \mapsvia{\sim} Y \fiberproduct{R} S^{\tensor_R 2} .\] - For any $G\dash$torsor $E$ and any $Y\in G\dash\Sch$ with an ample invertible $G\dash$linearized bundle[^g_linearized_bundle] , one can similarly define **twists** $\twistleft{E} Y$. - For $T$ a faithfully flat extension of $R$, the **Amitsur resolution** is given by \[ M\to M \tensor_R \TT(T)^\bullet && \text{where } \TT(V)^\bullet \da V \oplus V^{\tensor 2} \oplus \cdots .\] I.e., this resolves $M$ by the tensor algebra (or free algebra) on $T$. [^g_linearized_bundle]: This holds for example if $Y \in \Aff\Sch$. We'll now discuss some important special cases of $G\dash$torsors. The following claim is in analogy to $\Coh(X)$ for $X\in \Aff\Sch$: ::: :::{.fact title="Vector group schemes have trivial cohomology"} If $M\in \Fin\rmod^{\loc\Free, r<\infty}$, then $\cechH^1(R; W(M)) = 0$ and every $W(M)\dash$torsor is trivial. The following are some important special cases: ::: :::{.definition title="Finite Constant Group Scheme"} For any $\Gamma\in \Fin\Grp$, $\Gamma_R$ is the **finite constant group scheme** attached to any $\Gamma$, and is defined by \[ \Gamma_R(S) \da \ts{ f:\spec S\to \Gamma \st f\text{ is locally constant} } \in \Grp .\] ::: ## Galois Covers :::{.definition title="Galois Cover"} A $\Gamma_R\dash$torsor of the form $\spec S\to \spec R$ is equivalently a Galois $\Gamma\dash$algebra $S$ and is referred to as a **Galois cover**. ::: :::{.example title="?"} A finite Galois extension $L\slice{k}$ with Galois group $\Gamma \da \Gal(L\slice{k})$ yields a Galois cover $\spec L\to \spec k$. ::: :::{.example title="?"} Another nice case is when $\Gamma$ is the automorphism group associated to some algebraic structure, i.e. when one has **forms**. For example, take $\Gamma \da \Orth_{2n} = \Aut(q_\hyp)$ for the hyperbolic form $q_\hyp$ on $R^n$. Descent gives an equivalence of categories \[ \Grpd \ni \correspond{ \text{Regular quadratic forms } q \\ \text{with } \rank q = 2n} \mapstofrom H^1_{\fppf}(R; \Orth_{2n}) ,\] which uses that all forms appearing on the left-hand side are locally isomorphic to $q_\hyp$ in the flat topology. ::: :::{.example title="?"} Take $\Gamma \da S_n$ the symmetric group, so \[ S_n(X) = \Aut_\Grp(X^{\times n}) && \forall S \in \Alg_{/R} .\] The same yoga shows there is a categorical equivalence \[ S_n\dash\text{torsors} &\mapstofrom \Fin\Alg^{\et}_{/R} ,\] where we use that every $X\in \Fin\Alg^{\et}$ of degree $n$ is locally isomorphism to $R^n$. The inverse is given by descent. ::: ## Flat Quotients :::{.definition title="Flat Quotient"} For $X, H\in \Grp\Sch_{/R}$, a map $H\to X$ is a **flat quotient** of $H$ by $G$ iff - For each $S\in \ralg$ the map $H(S)\to X(S)$ induces an injection $H(S)/G(S)\injects X(S)$, and - For each $x\in X(S)$ there exists a flat cover $S'\to S$ with $x_{S'} \in \im(H(S') \to X(S'))$ (so $f$ is *couvrant* en français). ::: :::{.example title="of flat quotients"} \envlist - $\GG_m = \GL_r /_{\Flat}\, \SL_r$. - $\GG_m = \GG_m /_{\Flat}\, \mu_d$. ::: :::{.lemma title="?"} Let $X$ be the flat quotient of $H$ by $G$. 1. $H\to X$ is a $G\dash$torsor 2. There is an exact sequence of pointed sets: \[ 1 \to G(R) \to H(R) \to X(R) \mapsvia{\phi(x) = [f\inv(x)]} H_\fppf^1(R; G) \to H_\fppf^1(R; H) ,\] where $\phi$ is denoted the *characteristic map*, which arises naturally from base change. ::: :::{.remark} When $G\normal H$, then $X\in \Grp\Sch_{/R}$, and \[ \correspond{G\dash\text{torsors over }\spec R } \mapstofrom \correspond{(F, \phi) \st F\in H\dash\text{torsors}, \\ \phi \text{ a local trivialization of } \twistleft{F}X } .\] ::: :::{.example title="?"} $\SL_n\dash$torsors are equivalent to $(M, \theta)$ with $M\in \rmod^{\loc\Free, r}$ and $\theta: R \mapsvia{\sim} \Extalg^r(M)$ is a trivialization of $\det(M)$. ::: :::{.example title="?"} Using the Kummer exact sequence \[ 1 \to \mu_d \to \GG_m \mapsvia{\times d} \GG_m \to 1 ,\] $\mu_d\dash$torsors are equivalent to pairs $(M, \theta)$ with $M\in \rmod\units$ and $\theta: R \mapsvia{\sim} M^{\tensor_R m}$ is a trivialization. ::: :::{.definition title="étale morphism"} An **étale morphism** of rings $R\to S$ is a smooth morphism of $\reldim = 0$. Alternatively, $S\in \rmod^{\Flat}$ such that for every $R\dash$field $F$, $S\tensor_R F \in \Alg_{/F}^{\et}$, where étale algebras are finite and geometrically reduced. ::: :::{.example title="?"} - Localization $R\to R\localize{f}$ is étale. - If $d\inv \in R$ then the Kummer morphism $t\mapsto t^d$ is étale. - If $d\inv\in R$ and $r\in R\units$, then $R[x] /\gens{x^d-r}\in \Fin\Alg_{/R}^\et$. ::: :::{.proposition title="?"} For $G$ affine smooth, there is an equivalence of torsors \[ H^1_\et(R; G) \cong H^1_\fppf(R;G) .\] ::: :::{.proof title="?"} See notes. This uses that in the flat topology, smoothness is local. ::: ## Galois Cohomology :::{.remark} Galois descent is a special case of faithfully flat descent, and takes the form \[ \rmod \mapstofrom \correspond{(N, \rho) \st N\in \mods{S} \\ \phi \text{ a semilinear action } \Gamma\actson N } ,\] where **semilinear** means that \[ \varphi(\sigma) (\lambda \cdot n) = \sigma(\lambda) \cdot \rho( \sigma)(n) .\] ::: :::{.definition title="Isotrivial torsors (very important!)"} A torsor $X$ over $\Grp\Sch_{/R}$ is **isotrivial** if it is split (trivialized) by a finite Galois étale cover $R'\to R$. ::: :::{.remark} These are the torsors that can be made explicit in Galois cohomology. ::: :::{.example title="?"} Let $\ch k=0$, then the ring of Laurent polynomials $k\laurent{x}$, this is isotrivial and a reductive group scheme. A special case is that of **loop torsors**, which are closely related to representation theory in $\Alg\Grp$. ::: :::{.remark} The main topic is affine curves, and these are special cases of Dedekind rings. Let $R$ be Dedekind with $K\da \ff(R)$ and $G\in \Aff\Grp\Sch$, then as in the proof for $\GL_n$ yesterday we have an injective class map \[ \ker(H^1_\fppf \to H^1_\fppf \times \cdots ) \to c_{\Sigma}(\cdots) .\] In particular, if $c_\Sigma(R; G) =1$, and in particular $G(R\localize{f})$ is dense in $\prod \cdots$, the kernel appearing here is trivial. ::: :::{.corollary title="?"} If $G$ is a semisimple simply connected [^simply_connected_note] and split in $\Grp\Sch_{/R}$, then \[ \ker \qty{ H^1(R;G) \to H^1\qty{R\localize{f}; G}} = 1 .\] [^simply_connected_note]: Here "simply connected" is in the sense of semisimple algebraic groups or group schemes, and over $\CC$ coincides with the topological notion. ::: :::{.remark} This simplification comes from the injectivity of the following: \[ H^1( \hat R\localize{\mfp_i}; G) \injects H^1(\hat K \localize{\mfp_i}; R) .\] In the limit, this says that many torsors are actually trivial. We find that $H^1_\zar(R; \SL_n) = 1$ and $H^1_\zar(R; E_8) = 1$. ::: ## Curves Over an Algebraically Closed Field :::{.theorem title="?"} Let $k = \bar{k}$ and $G \in \Alg\Grp_{\slice{k}}$ be semisimple and $C$ a smooth connected curve. Then $H^1_\fppf(C; G) = 1$. ::: :::{.remark} The main ingredient is Steinberg's theorem that $H^1(K(C); G) = 1$. A special case is $\PGL_n$ and rephrases that central simple algebras over $k(C)$ are matrix algebras (using Tsen's theorem). This also uses that $\Pic(C)$ is divisible, which follows from the structure of $\Pic(C^c)$ for $C^c$ a smooth compactification. ::: :::{.remark} We have a degree map and moreover an exact sequence involving the Jacobian (an abelian variety): \[ 0 \to J_{C^c}(k) \to \Pic(C^c) \mapsvia{\deg} \ZZ \to 0 .\] If $C$ is $C^c$ minus finitely many points, $\Pic(C^c)\surjects \Pic(C)$ induces $J_{C^c}(k) \surjects \Pic(C)$. 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. $\SL_n\to \PGL_n$ or $\Spin_n\to \SO_n$. Let $\tilde T$ be its maximal torus, then $T\da \tilde T/\ker f$ is a maximal torus in $G$, so let $B \subseteq G$ be a Borel containing $T$. :::{.claim} \[ H^1(C; B) \surjects H^1(C; G) .\] ::: Letting $E$ be a $C\dash$torsor under $G$, then the idea is to introduce the twisted $C\dash$scheme $Y\da E(G/B)$, 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 $C\dash$point, so $E(G)$ has a Borel subgroup scheme. By functoriality, \[ [E] \in \im(H^1(C; B) \to H^1(C; G)) .\] We thus have $B = U\semidirect T$ where $U$ admits a $T\dash$equivariant filtration with associated quotients isomorphic to copies of $\GG_a$, and we apply a dévissage argument. Since $\tilde T\to T$ is an isogeny (finite kernel) and $\Pic(C)$ is a divisible group, a commutative diagram shows surjectivity $H^1(C; T)\surjects H^1(C; G)$ and thus the latter is trivial. The reductive case is similar, letting $S = G/DG$ be the coradial torus of $G$ and showing $H^1(C; G) \mapsvia{\sim} H^1(C; S)$ generalizing the bijection from yesterday: \[ H^1(C; \GL_r) \to H^1(C; \GG_m) = \Pic(C) .\] ::: ## Affine Line :::{.theorem title="?"} Let $k \in \Field$ be not necessarily algebraically closed and $G\in\Grp\Sch_{\slice{k}}$ reductive. We have a bijection \[ H^1(k; G) \mapsvia{\sim} \ker\qty{ H^1(k[t]; G) \to H^1(k_s[t]; G)} .\] If $k$ is perfect or $\ch k = p$ 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 $G\dash$torsors over $k[t]$ are **constant**. ::: :::{.remark} This doesn't hold for $G = \PGL_p$ over $k(t)$ with $\ch k = p$ imperfect. Our next goal is to prove this theorem -- a common ingredient to all proofs is the following theorem on bundles over $\PP^1$: ::: :::{.theorem title="Grothendieck-Harder"} For $G$ a reductive $k\dash$group, let $S$ be a maximal $k\dash$split torus of $G$ and consider its constant associated Weyl group \[ W_G(S) = N_G(S) / C_G(S) .\] Then there is a bijection \[ H^1_\zar(\PP^1_{\slice{k}}; S)/W_G(S) \mapsvia{\sim} \ker\qty{H^1(\PP^1_{\slice{k}}; G) \mapsvia{\ev_0} H^1(k; C)} .\] In particular if a $G\dash$torsor over $k[t]$ is trivial at $t=0$ and extends to a $G\dash$torsor over $\PP^1_{\slice{k}}$, then it is trivial. ::: :::{.remark} Given a $G\dash$torsor over $k[t]$, without loss of generality, we can assume $X$ is trivial on $t=0$ -- the original method to extend $X$ to $\PP^1$ is to use Bruhat-Tits theory. The idea is to find an integer $d\geq 1$ where $\gamma_{k[t^d]}$ extends to $\PP^1$. 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: \[ H^1(k((t\inv)); S)\to H^1(k((t\inv)); G) ,\] and we can write the absolute Galois group of $k\laurent{t\inv}$ as \[ \lim_n \mu_n(k_s) \semidirect G(k_s{}\slice{k}) = I \semidirect G(k_s {}\slice{k}) .\] 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 $F \da k(u)$. The upshot is that \[ \ff \qty{ k(u, t, (ut)^{1\over d}) } \cong k(t,x) .\] by a $k(t)\dash$linear isomorphism. The kernel is trivial by a specialization argument, so $\gamma$ is rationally trivial and extends to infinity. ::: :::{.remark} Noting that $\AA^1 = \PP^1 \smts{\pt}$, we have $\GG_m = \PP^1\smts{\pt_0, \pt_1}$, which is much more difficult. ::: :::{.theorem title="?"} Let $G$ be a reductive $k\dash$group over $\ch k = 0$, then there is a bijection \[ H^1(k[t, t\inv ]; G) \mapsvia{\sim} H^1(k\laurent{t}; G) .\] ::: :::{.remark} Surjectivity is easy, coming from reduction to a finite subgroup, and injectivity is hard. A crucial step is to show existence of a maximal torus for the relevant twisted group scheme, using again Bruhat-Tits theory and now *twin buildings*. So we have a good understanding of $\GG_m\dash$torsors, and a next step would be understanding $\PP^n$ with more deleted points. :::