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March 2014 A dual to tight closure theory
Neil Epstein, Karl Schwede
Nagoya Math. J. 213: 41-75 (March 2014). DOI: 10.1215/00277630-2376749

Abstract

We introduce an operation on modules over an F-finite ring of characteristic p. We call this operation tight interior. While it exists more generally, in some cases this operation is equivalent to the Matlis dual of tight closure. Moreover, the interior of the ring itself is simply the big test ideal. We directly prove, without appeal to tight closure, results analogous to persistence, colon capturing, and working modulo minimal primes, and we begin to develop a theory dual to phantom homology. Using our dual notion of persistence, we obtain new and interesting transformation rules for tight interiors, and so in particular for the test ideal. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory with Blickle’s notion of Cartier modules, and in the process we prove new existence results for Blickle’s test submodule. Finally, we apply the theory we developed to the study of test ideals in nonnormal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.

Citation

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Neil Epstein. Karl Schwede. "A dual to tight closure theory." Nagoya Math. J. 213 41 - 75, March 2014. https://doi.org/10.1215/00277630-2376749

Information

Published: March 2014
First available in Project Euclid: 31 October 2013

zbMATH: 1315.13016
MathSciNet: MR3290685
Digital Object Identifier: 10.1215/00277630-2376749

Subjects:
Primary: 13A35
Secondary: 13B22, 13B40, 14B05, 14F18

Rights: Copyright © 2014 Editorial Board, Nagoya Mathematical Journal

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Vol.213 • March 2014
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