Cohen-Macaulay test ideals over rings of finite and countable Cohen-Macaulay type

TL;DR

This paper investigates the computation of test ideals in rings with finite and countable Cohen-Macaulay type, linking module closures and trace ideals, and simplifies calculations for certain hypersurface rings.

Contribution

It provides new methods for computing test ideals using module trace ideals and offers simplified procedures for specific hypersurface rings.

Findings

Test ideals of rings with Cohen-Macaulay modules are computed using trace ideals.

An easier method is introduced for calculating test ideals in 3-variable hypersurface rings.

The relationship between module closures and test ideals is clarified.

Abstract

The third named author and P\'{e}rez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the closures coming from their indecomposable maximal Cohen-Macaulay modules. We also give an easier way to compute the test ideal of a hypersurface ring in 3 variables coming from a module with a particular type of matrix factorization.