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Transactions of the American Mathematical Society

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

 

 

Test ideals in quotients
of $F$-finite regular local rings


Author: Janet Cowden Vassilev
Journal: Trans. Amer. Math. Soc. 350 (1998), 4041-4051
MSC (1991): Primary 13A35
DOI: https://doi.org/10.1090/S0002-9947-98-02128-X
MathSciNet review: 1458336
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Abstract: Let $S$ be an $F$-finite regular local ring and $I$ an ideal contained in $S$. Let $R=S/I$. Fedder proved that $R$ is $F$-pure if and only if $(I^{[p]}:I)\break \nsubseteq \mathfrak{m}^{[p]}$. We have noted a new proof for his criterion, along with showing that $(I^{[q]}:I) \subseteq (\tau ^{[q]}:\tau )$, where $\tau $ is the pullback of the test ideal for $R$. Combining the the $F$-purity criterion and the above result we see that if $R=S/I$ is $F$-pure then $R/\tau $ is also $F$-pure. In fact, we can form a filtration of $R$, $I \subseteq \tau = \tau _{0} \subseteq \tau _{1} \subseteq \ldots \subseteq \tau _{i} \subseteq \ldots $ that stabilizes such that each $R/\tau _{i}$ is $F$-pure and its test ideal is $\tau _{i+1}$. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let $R=T/I$, where $T$ is either a polynomial or a power series ring and $I= P_{1} \cap \ldots \cap P_{n}$ is generated by monomials and the $R/P_{i}$ are regular. Set $J = \Sigma (P_{1} \cap \ldots \cap \hat {P_{i}} \cap \ldots \cap P_{n})$. Then $J=\tau =\tau _{par}$.


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Additional Information

Janet Cowden Vassilev
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90024
Address at time of publication: Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, Virginia 23284
Email: jcvassil@saturn.vcu.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02128-X
Keywords: Tight closure, test element, $F$-finite, $F$-pure
Received by editor(s): November 4, 1996
Additional Notes: I would like to express my appreciation to Purdue University for hosting me during the time that I completed these results. I also thank Craig Huneke for many helpful conversations.
Article copyright: © Copyright 1998 American Mathematical Society