Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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

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


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 $J=\tau =\tau _{par}$.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13A35
  • Retrieve articles in all journals with MSC (1991): 13A35
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:
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4041-4051
  • MSC (1991): Primary 13A35
  • DOI:
  • MathSciNet review: 1458336