# Monday April 27th ## Principle of Galois Cohomology Let $\ell_{/k}$ a galois extension and $X_{/k}$ some "object" for which it makes sense to associate another object over $\ell$. We'll prove that there's a correspondence \[ \correspond{ \ell_{/k}, \text{ twisted forms} \\ Y \text{ of } X_{/k} } &\mapstofrom H^1(\ell_{/k}, \aut(X_{/\ell})) .\] Recall that $\PGL(n ,\ell) \da \GL(n ,\ell) / \ell\units$. :::{.example title="?"} Let $X = \PP^{n-1}/k$, then $H^1(\ell_{/k}, \PGL(n, \ell)$ parameterizes twisted forms of $\PP^{n-1}$, e.g. for $n=2$ twisted forms of $\PP^1$ and plane curves. ::: :::{.example title="?"} Take $X = M_{n}(k)$ the algebra of $n\times n$ matrices. Then by a theorem (Skolern-Noether) $\aut(M_{n}(k)) = \PGL(n, k)$. Thus $H^1(\ell_{/k}, \PGL(n, k))$ also parameterizes twisted forms of $M_{n}(k)$ in the category of unital (not necessarily commutative) $k\dash$algebras. These are exactly central simple algebras $A_{/k}$ where $\dim_{k} A = n^2$ with center $Z(A) = k$ with no nontrivial two-sided ideals. By taking \(\ell = k^{s} \), we get a correspondence \[ \correspond{\text{CSAs} A_{/k} \text{ of degree } n} &\mapstofrom \correspond{\text{ Severi-Brauer varieties of dimension n-1} } .\] Taking $n=2$ we obtain \[ \correspond{\text{Quaternion algebras } A_{/k}} &\mapstofrom \correspond{\text{Genus 0 curves } \ell_{/k}} .\] ::: ## The Weil Descent Criterion Fix $\ell_{/k}$ finite Galois with $g \da \aut(\ell_{/k})$. 1. $X_{/k} \to X_{/\ell}$ with a $g\dash$action. 2. What additional data on an $\ell\dash$variety $Y_{/\ell}$ do we need in order to "descend the base" from $\ell$ to $k$? For $\sigma \in g$, write $\ell^\sigma$ to denote $\ell$ given the structure of an $\ell\dash$algebra via $\sigma: \ell \to \ell^\sigma$. If $X_{/\ell}$ is a variety, so is $X^\sigma_{/\ell}$? \begin{tikzcd} X^\sigma\ar[dr, dotted] \ar[r]\ar[d] & X \ar[d] \\ \spec \ell^\sigma \ar[r, "f"] & \spec \ell \end{tikzcd} where $f$ is the map induced on $\spec$ by $\sigma$. We can also think of these on defining equations: \[ X &= \spec \ell[t_{1}, \cdots, t_{n}] / \gens{p_{1}, \cdots, p^n} \\ X^\sigma &= \spec \ell[t_{1}, \cdots, t_{n}] / \gens{\sigma_{p_{1}}, \cdots, \sigma{p^n}} \\ .\] For $X_{/k}, X_{/\ell}$, we canonically identify $X$ with $X^\sigma$ by the map $f_\sigma: X \mapsvia{\cong} X^\sigma$, a canonical isomorphism of $\ell\dash$varieties. We thus have \begin{tikzcd} X \ar[r, "f_\sigma"] \ar[rr, bend left, "f_{\sigma \tau}"] & X^\sigma \ar[r, "f_\sigma"] & X^{\sigma \tau} \end{tikzcd} under a "cocycle condition" $f_{\sigma \tau} = {}^\sigma f_\tau \circ f_\sigma$. :::{.theorem title="Weil"} Given $Y_{/\ell}$ quasi-projective and $\forall \sigma \in g$ we have descent datum $f_\sigma: Y\mapsvia{\cong} Y^\sigma$ satisfying the above cocycle condition, and there exists a unique $X_{/k}$ such that $X_{/\ell} \mapsvia{\cong} Y_{/\ell}$ and the descent data coincide. ::: ### An Application Let $X_{/k}$ be a quasiprojective variety and $Y_{/k}$ and $\ell_{/k}$ twisted forms. Then $a_{0} \in Z' (\ell_{/k}, \aut X)$. Conversely, we have the following: :::{.definition title="Twisted Descent Data"} Let $a_{0}$ be such a cocycle and $\theset{s_\sigma: X\to X^\sigma}$ be descent datum attached to $X$. Define twisted descent datum $g_\sigma \da f_\sigma \circ a_\sigma$ from \[ X /\ell\mapsvia{a_\sigma} X_{/\ell} \mapsvia{f_\sigma} X^\sigma / \ell .\] ::: :::{.exercise title="?"} Check that $g_\sigma$ satisfies the cocycle condition, so by Weil uniquely determines a ($k\dash$model) $Y_{/k}$ of $X_{/\ell}$. ::: :::{.example title="?"} Let $G_{/k}$ be a smooth algebraic group and $X_{/k}$ a torsor under $G$. Then $\Aut(G) \supset \aut_{G\dash\text{torsor}} (G) = G$, since in general the translations will only be a subgroup of the full group of automorphisms. Then \[ H^1(\ell_{/k}, G) \to H^1(\ell_{/k}, \aut G) \] defines a twisted form $X$ of $G$. How do you descend the torsor structure? This is possible, but not covered in Bjoern's book! This requires expressing the descent data more functorially -- see the book on Neron models. ::: ## The Cohomology Theory ### Motivation Let $G_{/k}$ be a smooth connected commutative algebraic group where $\ch k$ does not divide $n$, so the map $[n]: G \to G$ is an isogeny. Then \[ 0 \to G[n] (k^{s} ) \to G(k^{s} ) \mapsvia{[n]} G(k^{s} ) \to 0 \] is a SES of $g = \aut(k^{s}_{/k})\dash$modules. :::{.claim} Taking the associated cohomology sequence yields the Kummer sequence: \[ 0 \to G(k) / nG(k) \to H^1(k, G[n]) \to H^1(k, G)[n] \to 0 \] where the RHS is the **Weil-Châtelet** group and the LHS is the **Mordell-Weil** group. ::: For $g$ a profinite group, a commutative discrete $g\dash$group is by definition a $g\dash$module. These form an abelian category with enough injectives, so we can take right-derived functors of left-exact functors. We will consider the functor $$ A \mapsto A^g \da \theset{x\in A \suchthat \sigma x = x ~\forall \sigma\in g} ,$$ then define $H^i(g, A)$ to be the $i$th right-derived functor of $A \mapsto A^\sigma$. This is abstractly defined by taking an injective resolution, applying the functor, then taking cohomology. A concrete description is given by $C^n(g, A) = \Map(g^n, A)$ with \[ d: C^n(g, A) &\to C^{n+1}(g, A) \\ (df)(\sigma_{1}, \cdots, \sigma_{n+1} &\da \sigma_{1} f(\sigma_{2}, \cdots, \sigma_{n+1}) \\ &\qquad + \sum_{i=1}^n (-1) f(\sigma _1, \cdots, \sigma_{i-1}, \sigma_{i}, \sigma_{i+1}, \cdots, \sigma_{n+1}) \\ &\qquad + (-1)^{n+1} f(\sigma_{1}, \cdots, \sigma_{n}) .\] Then $d^2 = 0$, $H^n$ is kernels mod images, and this agrees with $H^1$ as defined before with $H^0 = A^g$. We'll see that that \[ H^i(g, A) = \directlim_{U} G^i(g/U, A^U) .\] If $g$ is finite, $A$ is a $g\dash$module $\iff$ $A$ is a $\ZZ[g]\dash$module, and thus \[ A^g = \hom_{\ZZ[g]\dash\text{mod}}(\ZZ, A) .\] where $\ZZ$ is equipped with a trivial $g\dash$action. We can thus think of \[ H^i(g, A) = \ext^i_{\ZZ[g]}(\ZZ, A) .\] > The end!