The cohomology ring of a finite group scheme
Proc. Amer. Math. Soc. 26 (1970), 567-570
Primary 14.50; Secondary 18.00
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Abstract: Let be a field and let be a -algebra with additional structure so that Spec is a finite commutative group scheme over , (so is a Hopf algebra). Let be the Hochschild cohomology ring. In another paper, we demonstrated that if is a perfect field:
(a) is generated by and .
(b) If characteristic , then is freely generated by and .
(c) If characteristic , then there are subspaces of and of such that is generated by and the only relations are for all in .
In this paper we show that if 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 .
Cartan and Samuel
Eilenberg, Homological algebra, Princeton University Press,
Princeton, N. J., 1956. MR 0077480
Efroymson, Certain cohomology rings of finite and
formal group schemes, Trans. Amer. Math.
Soc. 145 (1969),
309–322. MR 0258843
(41 #3489), http://dx.doi.org/10.1090/S0002-9947-1969-0258843-0
Oort and David
Mumford, Deformations and liftings of finite, commutative group
schemes, Invent. Math. 5 (1968), 317–334. MR 0228505
- H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
- G. Efroymson, Certain cohomology rings of finite and formal group schemes, Trans. Amer. Math. Soc. 145 (1969), 309-322. MR 0258843 (41:3489)
- F. Oort and D. Mumford, Deformations and liftings of finite commutative group schemes, Invent. Math. 5 (1968), 317-344. MR 37 #4085. MR 0228505 (37:4085)
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