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A dual to tight closure theory

Published online by Cambridge University Press:  11 January 2016

Neil Epstein and
Karl Schwede
Neil Epstein
Affiliation:
Department of Mathematical Sciences George Mason University, Fairfax, Virginia 22030, USA, nepstei2@gmu.edu
Karl Schwede
Affiliation:
Department of Mathematics The Pennsylvania State University, University Park, Pennsylvania 16802, USA, schwede@math.psu.edu
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Abstract

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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.

Keywords

13A3513B2213B4014B0514F18

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 213 , March 2014, pp. 41 - 75
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

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