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$F$-injective rings and $F$-stable primes

Author: Florian Enescu
Journal: Proc. Amer. Math. Soc. 131 (2003), 3379-3386
MSC (2000): Primary 13A35
Published electronically: March 25, 2003
MathSciNet review: 1990626
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Abstract: The notion of stability of the highest local cohomology module with respect to the Frobenius functor originates in the work of R. Hartshorne and R. Speiser. R. Fedder and K.-i. Watanabe examined this concept for isolated singularities by relating it to $F$-rationality. The purpose of this note is to study what happens in the case of non-isolated singularities and to show how this stability concept encapsulates a few of the subtleties of tight closure theory. Our study can be seen as a generalization of the work by Fedder and Watanabe. We introduce two new ring invariants, the $F$-stability number and the set of $F$-stable primes. We associate to every ideal $I$ generated by a system of parameters and $x \in I^\ast - I$ an ideal of multipliers denoted $I(x)$ and obtain a family of ideals $Z_{I,R}$. The set $\operatorname{Max}(Z_{I,R})$ is independent of $I$ and consists of finitely many prime ideals. It also equals $ \operatorname{Max} \{P\vert P $ prime ideal such that $ R_{P} $ is $F$-stable$ \}$. The maximal height of such primes defines the $F$-stability number.

References [Enhancements On Off] (What's this?)

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Additional Information

Florian Enescu
Affiliation: Department of Mathematics, University of Utah, 1400 East, 155 South, Salt Lake City, Utah 84112 – and – Institute of Mathematics of the Romanian Academy, Bucharest, Romania

Keywords: Tight closure, local cohomology
Received by editor(s): March 1, 2002
Received by editor(s) in revised form: June 14, 2002
Published electronically: March 25, 2003
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2003 American Mathematical Society