## Abstract

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 = G_{a, k}, let α_{pν}denote the subgroup of p^{ν}nilpotents and let F_{ν}, denote the νth power of the Frobenius. We show that for any finite M that there is a ν such that H•(G_{a, k}, M)∜ H•(α_{pν}, M)⊗_{k}F*_{ν}(H•(G_{a, k})) where F*_{ν} is the endomorphism of H•(G_{a, k}) induced by F_{ν}. As a consequence, we can show that H• (G_{a, k}, M) is a finitely generated module over H•(G_{a, k}) when M is a finite dimensional vector space over k.

Original language | English (US) |
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Pages (from-to) | 553-565 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 339 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1993 |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics