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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Principal homogeneous spaces and group scheme extensions

Author: William C. Waterhouse
Journal: Trans. Amer. Math. Soc. 153 (1971), 181-189
MSC: Primary 14.50; Secondary 13.00
MathSciNet review: 0269659
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Abstract: Suppose $ G$ is a finite commutative group scheme over a ring $ R$. Using Hopf-algebraic techniques, S. U. Chase has shown that the group of principal homogeneous spaces for $ G$ is isomorphic to $ \operatorname{Ext} (G',{G_m})$, where $ G'$ is the Cartier dual to $ G$ and the Ext is in a specially-chosen Grothendieck topology. The present paper proves that the sheaf $ \operatorname{Ext} (G',{G_m})$ vanishes, and from this derives a more general form of Chase's theorem. Our Ext will be in the usual (fpqc) topology, and we show why this gives the same group. We also give an explicit isomorphism and indicate how it is related to the existence of a normal basis.

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Keywords: Commutative group scheme, principal homogeneous space, line bundle, extension group, dual group scheme, Grothendieck topology, normal basis
Article copyright: © Copyright 1971 American Mathematical Society