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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0266928-1
MathSciNet review:
0266928
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Abstract | References | Similar Articles | Additional Information
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 $k$.
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- Gustave Efroymson, Certain cohomology rings of finite and formal group schemes, Trans. Amer. Math. Soc. 145 (1969), 309–322. MR 258843, DOI https://doi.org/10.1090/S0002-9947-1969-0258843-0
- Frans Oort and David Mumford, Deformations and liftings of finite, commutative group schemes, Invent. Math. 5 (1968), 317–334. MR 228505, DOI https://doi.org/10.1007/BF01389779
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Additional Information
Keywords:
Finite group scheme,
Hochschild cohomology ring,
<!– MATH $H_{\operatorname {sym} }^2(A,k)$ –> <IMG WIDTH="102" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$H_{\operatorname {sym} }^2(A,k)$">,
Hopf algebra
Article copyright:
© Copyright 1970
American Mathematical Society