Hochschild cohomology of algebraic groups and the classification of Hochschild extensions

Let $k$ be a field, let $G$ be an algebraic group over $k$ (Group schemes of finite type over a field) and let $M$ be a $G$-module: a commutative group functor on $k$-algebras equipped with a left action of $G$ by group homomorphisms. A typical example is a rational representation of an affine $G$ viewed as a group functor (Rational representations and comodules of an affine group scheme).

For $n\ge 0$ put $C^n(G,M)=\mathrm{Nat}(G^n,M)=\{\text{natural transformations of set-valued functors }{G}^n\to M\},$ the group of $n$-cochains, with pointwise addition. Here each cochain is a family of maps on algebra-valued points compatible with every base change, rather than an arbitrary function on one point set; for represented functors these are morphisms of schemes by Yoneda. With the convention $G^0=\mathrm{Spec}k$ so that $C^0(G,M)=M(k)$. The coboundary ${\mathrm{\partial }}^n:C^n(G,M)\to C^{n+1}(G,M)$ is $({\mathrm{\partial }}^{n}f)(g_1,\dots ,g_{n+1})=g_1\cdot f(g_2,\dots ,g_{n+1})+{\sum }_{i=1}^n(-1{)}^{i}f(g_1,\dots ,g_i{g}_{i+1},\dots ,g_{n+1})+(-1{)}^{n+1}f(g_1,\dots ,g_n),$ evaluated functorially on $k$-algebras; one checks ${\mathrm{\partial }}^{n+1}{\mathrm{\partial }}^n=0$ by the usual alternating-sum cancellation, using functoriality of the $G$-action on $M$. The Hochschild cohomology of $G$ with coefficients in $M$ is $H^n(G,M)=\ker {\mathrm{\partial }}^n\mathrm{/}\mathrm{im}{\mathrm{\partial }}^{n-1},\mathrm{im}{\mathrm{\partial }}^{-1}:=0,$ so in particular $H^0(G,M)=M(k{)}^G$ is the group of $G$-invariant $k$-points, and $C^{\bullet }(G,M)$ is a complex of abelian groups.

An exact sequence of $G$-modules $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$ gives a long exact sequence when its induced cochain maps $C^n(G,M)\to C^n(G,M^{\mathrm{\prime }\mathrm{\prime }})$ are surjective in every degree (for example when $M\to M^{\mathrm{\prime }\mathrm{\prime }}$ has a section as a map of set-valued functors). Then the induced complexes form a short exact sequence, and the usual connecting-map construction gives $\cdots \to H^n(G,M^{\mathrm{\prime }})\to H^n(G,M)\to H^n(G,M^{\mathrm{\prime }\mathrm{\prime }})\stackrel{\delta }{\to }{H}^{n+1}(G,M^{\mathrm{\prime }})\to \dots$

For rational modules $V$, this surjectivity always holds for an exact sequence of representations: any $k$-linear splitting of the coefficient vector spaces is a natural map of their additive functors, and applying it to a cochain gives a lift (equivariance of that splitting is not required). If $G$ is affine with coordinate ring $A$, Yoneda gives $C^n(G,V^{\mathrm{a}})=V{\otimes }_k{A}^{\otimes n}$, so degreewise exactness also follows directly from tensoring vector spaces over a field. These rational cochains commute with filtered unions of coefficient submodules, because each tensor is a finite sum. The rational-module statements on this page use this case.

For the second cohomology group the following classification holds (Crossed homomorphisms, principal crossed homomorphisms and Hochschild extensions for the terminology). Let $E(G,M)$ be the set of equivalence classes of Hochschild extensions $0\to M\to E\to G\to 1$ inducing the given action of $G$ on $M$, two extensions being equivalent when they are isomorphic over $G$ by a map restricting to the identity on $M$. Then there is a canonical bijection $E(G,M)⟶H^2(G,M)$ sending the class of an extension with a section $s:G\to E$ to the class of the $2$-cocycle $f(g_1,g_2)=s(g_1)s(g_2)s(g_1{g}_2{)}^{-1}\in M,$ i.e. the unique element with $s(g_1)s(g_2)=f(g_1,g_2)s(g_1{g}_2)$; the class of $f$ is independent of the choice of section, and a $2$-cocycle conversely determines an extension with the given action. This is Milne's Proposition 15.10; the definitions and the functoriality statements used here are the formal parts of Sections 15(b)-(c) of the cited source.