A common framework for test ideals, closure operations, and their duals
Neil Epstein, Rebecca R. G., Janet Vassilev
TL;DR
This survey unifies closure and interior theories in commutative algebra through the framework of pair operations and a smile dual induced by Matlis duality. It shows how to systematically convert and compare closures, interiors, and their duals, and develops finitistic, residual, cohereditary, and hereditary variants, including connections to submodule selectors and pre-enveloping classes. The framework clarifies how standard constructions such as tight closure, integral closure, test ideals, traces, and core/hull fit into a common lattice-theoretic and duality-driven theory, yielding concrete formulas and transfer principles across contexts. Overall, the work provides a versatile toolbox for deriving dualities and compatibility results among a broad family of module-closure-type operations, with implications for singularity theory and algebraic geometry.
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.