Annihilating the cohomology of group schemes

Annihilating the Cohomology of Group Schemes

Our goal in this note is to show that cohomology classes with coefficients in finite flat group schemes can be killed by finite covers of the base scheme, and similarly for abelian schemes with “finite covers” replaced by “proper covers.” We apply this result to commutative algebra by giving a new and more conceptual proof of Hochster-Huneke’s theorem on the existence of big Cohen-Macaulay alge...

Module cohomology group of inverse semigroup algebras

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

Cohomology of Finite Group Schemes over a Field

A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]-comodule M . This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] with coefficients in the k[G]-module ...

Cohomology for Drinfeld Doubles of Some Infinitesimal Group Schemes

Consider a field k of characteristic p > 0, G(r) the r-th Frobenius kernel of a smooth algebraic group G, DG(r) the Drinfeld double of G(r), and M a finite dimensional DG(r)-module. We prove that the cohomology algebra H(DG(r), k) is finitely generated and that H(DG(r),M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θr : H(G(r), k) ⊗ S(g) → H(...

Cohomology of Schemes