Frobenius maps on injective hulls and their applications to tight closure

TL;DR

This paper explores Frobenius maps on injective hulls in complete local rings to develop algorithms for computing tight closure and test ideals, enhancing constructive methods in tight closure theory.

Contribution

It introduces algorithms for computing parameter test ideals and tight closure in a broad class of rings, including quasi-Gorenstein rings, advancing practical applications of tight closure.

Findings

Algorithms for parameter test ideals

Algorithms for tight closure of submodules

Applicable to quasi-Gorenstein rings

Abstract

This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view toward providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved rings which includes all quasi-Gorenstein complete local rings.