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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The cohomology ring of a finite group scheme


Author: Gustave Efroymson
Journal: Proc. Amer. Math. Soc. 26 (1970), 567-570
MSC: Primary 14.50; Secondary 18.00
MathSciNet review: 0266928
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Abstract: Let $ k$ be a field and let $ A$ be a $ k$-algebra with additional structure so that Spec $ A$ is a finite commutative group scheme over $ k$, (so $ A$ is a Hopf algebra). Let $ H^\bullet(A,k)$ be the Hochschild cohomology ring. In another paper, we demonstrated that if $ k$ is a perfect field:

(a) $ H^\bullet(A,k)$ is generated by $ {H^1}$ and $ H_{\operatorname{sym} }^2$.

(b) If characteristic $ k = p \ne 2$, then $ H^\bullet(A,k)$ is freely generated by $ {H^1}$ and $ H_{\operatorname{sym} }^2$.

(c) If characteristic $ k = 2$, then there are subspaces $ {V_1},{V_2}$ of $ {H^1}$ and $ {V_3}$ of $ H_{\operatorname{sym} }^2$ such that $ H^\bullet(A,k)$ is generated by $ {V_1},{V_2},{V_3}$ and the only relations are $ {f^2} = 0$ for all $ f$ in $ {V_1}$.

In this paper we show that if $ k$ is arbitrary (a) and (b) still hold, and we use an example of Oort and Mumford to show that (c) does not hold for arbitrary $ k$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0266928-1
PII: S 0002-9939(1970)0266928-1
Keywords: Finite group scheme, Hochschild cohomology ring, $ H_{\operatorname{sym} }^2(A,k)$, Hopf algebra
Article copyright: © Copyright 1970 American Mathematical Society