# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Test ideals in quotients of $F$-finite regular local ringsHTML articles powered by AMS MathViewer

by Janet Cowden Vassilev
Trans. Amer. Math. Soc. 350 (1998), 4041-4051 Request permission

## 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 $(I^{[p]}:I) \nsubseteq \mathfrak {m}^{[p]}$. 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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