A dual to tight closure theory

TL;DR

This paper introduces the concept of tight interior as a dual operation to tight closure in modules over F-finite rings, providing new insights and results in test ideal theory without relying on tight closure.

Contribution

It develops a dual theory to tight closure called tight interior, establishes new properties and transformation rules, and applies these to test ideals in non-normal rings.

Findings

Tight interior is equivalent to the Matlis dual of tight closure in some cases.

New vanishing theorem for Ext maps using phantom homology.

Finitistic and big test ideals coincide in certain non-normal rings.

Abstract

We introduce an operation on modules over an $F$-finite ring of characteristic $p$. We call this operation \emph{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 interior, and so in particular for the test ideal, which complement the main results of a recent paper of the second author and K. Tucker. Using our theory of phantom homology, we prove a vanishing theorem for maps of Ext. We also compare our theory to M. 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 non-normal rings, proving that the finitistic test ideal coincides with the big test ideal in some cases.