## Nagoya Mathematical Journal

### A dual to tight closure theory

#### 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.

#### Article information

Source
Nagoya Math. J., Volume 213 (2014), 41-75.

Dates
First available in Project Euclid: 31 October 2013

https://projecteuclid.org/euclid.nmj/1383240639

Digital Object Identifier
doi:10.1215/00277630-2376749

Mathematical Reviews number (MathSciNet)
MR3290685

Zentralblatt MATH identifier
1315.13016

#### Citation

Epstein, Neil; Schwede, Karl. A dual to tight closure theory. Nagoya Math. J. 213 (2014), 41--75. doi:10.1215/00277630-2376749. https://projecteuclid.org/euclid.nmj/1383240639

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