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Transactions of the American Mathematical Society

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



Test ideals in local rings

Author: Karen E. Smith
Journal: Trans. Amer. Math. Soc. 347 (1995), 3453-3472
MSC: Primary 13A35; Secondary 13H10
MathSciNet review: 1311917
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Abstract: It is shown that certain aspects of the theory of tight closure are well behaved under localization. Let $ J$ be the parameter test ideal for $ R$, a complete local Cohen-Macaulay ring of positive prime characteristic. For any multiplicative system $ U \subset R$, it is shown that $ J{U^{ - 1}}R$ is the parameter test ideal for $ {U^{ - 1}}R$. This is proved by proving more general localization results for the here-introduced classes of " $ {\text{F}}$-ideals" of $ R$ and " $ {\text{F}}$-submodules of the canonical module" of $ R$, which are annihilators of $ R$ modules with an action of Frobenius. It also follows that the parameter test ideal cannot be contained in any parameter ideal of $ R$.

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Keywords: Tight closure, test ideals, Frobenius action
Article copyright: © Copyright 1995 American Mathematical Society

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