Vanishing of Hochschild Cohomology for Affine Group Schemes and Rigidity of Homomorphisms between Algebraic Groups

Let $k$ be an algebraically closed field. If $\bG$ is a linearly reductive $k$--group and $\bH$ is a smooth algebraic $k$--group, we establish a rigidity property for the set of group homomorphisms $\bG \to \bH$ up to the natural action of $\bH(k)$ 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.

2000 Mathematics Subject Classification: 20G05

Keywords and Phrases: Group schemes, representations, linearly reductive group, deformation theory.

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