## Nagoya Mathematical Journal

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

### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00277630-2376749

**Mathematical Reviews number (MathSciNet)**

MR3290685

**Zentralblatt MATH identifier**

1315.13016

**Subjects**

Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]

Secondary: 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 13B40: Étale and flat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25] 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14F18: Multiplier ideals

#### 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.