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Transactions of the American Mathematical Society

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

 

 

Tameness and local normal bases for objects of finite Hopf algebras


Authors: Lindsay N. Childs and Susan Hurley
Journal: Trans. Amer. Math. Soc. 298 (1986), 763-778
MSC: Primary 13B05; Secondary 13B15, 13E15, 16A24
DOI: https://doi.org/10.1090/S0002-9947-1986-0860392-3
MathSciNet review: 860392
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Abstract: Let $ R$ be a commutative ring, $ S$ an $ R$-algebra, $ H$ a Hopf $ R$algebra, both finitely generated and projective as $ R$-modules, and suppose $ S$ is an $ H$-object, so that $ {H^{\ast}} = {\operatorname{Hom} _R}(H,R)$ acts on $ S$ via a measuring. Let $ I$ be the space of left integrals of $ {H^{\ast}}$. We say $ S$ has normal basis if $ S \cong H$ as $ {H^{\ast}}$modules, and $ S$ has local normal bases if $ {S_p} \cong {H_p}$ as $ H_p^{\ast}$-modules for all prime ideals $ p$ of $ R$. When $ R$ is a perfect field, $ H$ is commutative and cocommutative, and certain obvious necessary conditions on $ S$ hold, then $ S$ has normal basis if and only if $ IS = R = {S^{{H^{\ast}}}}$. If $ R$ is a domain with quotient field $ K$, $ H$ is cocommutative, and $ L = S \otimes {}_RK$ has normal basis as $ ({H^{\ast}} \otimes K)$-module, then $ S$ has local normal bases if and only if $ IS = R = {S^{{H^{\ast}}}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0860392-3
Article copyright: © Copyright 1986 American Mathematical Society