The cohomology algebra of a commutative group scheme

Robert Fossum, William Haboush

Research output: Contribution to journalArticlepeer-review


Let k be a commutative ring with unit of characteristic p > 0 and let G = Spec(A) be an affine commutative group scheme over k. Let H•(G) be the graded Hochschild algebraic group cohomology algebra and, for M a rational G-module, let H·(G, M) denote the graded Hochschild cohomology H•(G)-module. We show that H•(G) is, in general, a graded Hopf algebra. When G = Ga, k, let αpνdenote the subgroup of pνnilpotents and let Fν, denote the νth power of the Frobenius. We show that for any finite M that there is a ν such that H•(Ga, k, M)∜ H•(αpν, M)⊗kF*ν(H•(Ga, k)) where F*ν is the endomorphism of H•(Ga, k) induced by Fν. As a consequence, we can show that H• (Ga, k, M) is a finitely generated module over H•(Ga, k) when M is a finite dimensional vector space over k.

Original languageEnglish (US)
Pages (from-to)553-565
Number of pages13
JournalTransactions of the American Mathematical Society
Issue number2
StatePublished - Oct 1993

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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