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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


The cohomology algebra of a commutative group scheme

Authors: Robert Fossum and William Haboush
Journal: Trans. Amer. Math. Soc. 339 (1993), 553-565
MSC: Primary 14L17; Secondary 14L15, 16E40
MathSciNet review: 1112374
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Abstract: Let $ k$ be a commutative ring with unit of characteristic $ p > 0$ and let $ G = \operatorname{Spec}(A)$ be an affine commutative group scheme over $ k$. Let $ {{\text{H}}^ \bullet }(G)$ be the graded Hochschild algebraic group cohomology algebra and, for $ M$ a rational $ G$-module, let $ {{\text{H}}^ \bullet }(G,M)$ denote the graded Hochschild cohomology $ {{\text{H}}^ \bullet }(G)$-module. We show that $ {{\text{H}}^ \bullet }(G)$ is, in general, a graded Hopf algebra. When $ G = {{\mathbf{G}}_{a,k}}$, let $ {\alpha _{{p^\nu }}}$ denote the subgroup of $ {p^\nu }$-nilpotents and let $ {{\text{F}}_\nu }$ denote the $ \nu$th power of the Frobenius. We show that for any finite $ M$ that there is a $ \nu $ such that

$\displaystyle {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}},M) \cong {{\text{H}}^... ...otimes _k}{\text{F}}_{\nu }^\ast ({{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}}))$

where $ {\text{F}}_\nu ^\ast$ is the endomorphism of $ {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}})$ induced by $ {F_v}$. As a consequence, we can show that $ {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}},M)$ is a finitely generated module over $ {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}})$ when $ M$ is a finite dimensional vector space over $ k$.

References [Enhancements On Off] (What's this?)

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Additional Information

PII: S 0002-9947(1993)1112374-8
Article copyright: © Copyright 1993 American Mathematical Society