A common framework for test ideals, closure operations, and their duals

TL;DR

This paper surveys a unified framework connecting various closure operations, test ideals, and their duals in commutative algebra, highlighting dualities and methods for constructing compatible pair operations.

Contribution

It introduces a general duality between closure and interior operations, extending known dualities and providing methods for constructing compatible pair operations.

Findings

Establishes a duality between closure and interior operations.

Provides methods for creating compatible pair operations.

Generalizes the concepts of core and its dual.

Abstract

Closure operations such as tight and integral closure and test ideals have appeared frequently in the study of commutative algebra. This articles serves as a survey of the authors' prior results connecting closure operations, test ideals, and interior operations via the more general structure of pair operations. Specifically, we describe a duality between closure and interior operations generalizing the duality between tight closure and its test ideal, provide methods for creating pair operations that are compatible with taking quotient modules or submodules, and describe a generalization of core and its dual. Throughout, we discuss how these ideas connect to common constructions in commutative algebra.