Crossed homomorphisms, principal crossed homomorphisms and Hochschild extensions

Let $k$ be a field, let $G$ be an affine algebraic group over $k$ (Affine schemes and their coordinate rings, Group schemes of finite type over a field) and let $M$ be a commutative affine algebraic group over $k$ on which $G$ acts by group automorphisms, $\cdot :G{\times }_{k}M\to M$ (Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers). Form the semidirect product $M⋊G$, the $k$-group scheme whose $R$-points are the pairs $(m,g)\in M(R)\times G(R)$ with multiplication $(m,g)(m^{\mathrm{\prime }},g^{\mathrm{\prime }})=(m+g\cdot m^{\mathrm{\prime }},g{g}^{\mathrm{\prime }});$ the product is a group law because the action is by group automorphisms, and $M⋊G$ is constructed from the given morphisms using fibre products (Fibre product of schemes).

A crossed homomorphism is a morphism of $k$-schemes $f:G\to M$ such that $f(xy)=f(x)+x\cdot f(y)$ for all $x,y\in G(R)$ and all $k$-algebras $R$. It is principal if there is an element $m\in M(k)$ with $f(x)=x\cdot m-m\text{for all }x\in G(R),R\mathrm{.}$

The assignments $x\mapsto (f(x),x)$ and $(m,x)\mapsto x$ identify sections $G\to M⋊G$ of the projection $M⋊G\to G$ with crossed homomorphisms: indeed $(f(x),x)(f(y),y)=(f(x)+x\cdot f(y),xy)$, so multiplicativity of the section is exactly the displayed identity. Conjugating a section by $m\in M(k)$ changes the corresponding crossed homomorphism by the principal crossed homomorphism $x\mapsto x\cdot m-m$; hence two sections are conjugate by an element of $M(k)$ if and only if their crossed homomorphisms differ by a principal one.

An extension of group functors $0\to M\to E\to G\to 1$ is a sequence of group functors on $k$-algebras that is exact, with $E\to G$ the given projection. Such an extension is a Hochschild extension if the projection $E\to G$ admits a section as a map of set-valued functors. For a Hochschild extension the conjugation action of $G$ on $M$ induced by any such section is independent of the choice of section, and the equivalence classes of Hochschild extensions inducing a given action are classified by the second Hochschild cohomology group of Hochschild cohomology of algebraic groups and the classification of Hochschild extensions.