Vanishing of Hochschild cohomology for affine group schemes and rigidity of homomorphisms between algebraic groups

Abstract

Let $k$ be an algebraically closed field. If $\mathbf{G}$ is a linearly reductive $k$-group and $\mathbf{H}$ is a smooth algebraic $k$-group, we establish a rigidity property for the set of group homomorphisms $\mathbf{G}\to \mathbf{H}$ up to the natural action of $\mathbf{H}(k)$ by conjugation. Our main result states that this set remains constant under any base change $K/k$ with $K$ algebraically closed. This is proven as consequence of a vanishing result for Hochschild cohomology of affine group schemes.