Closure operations that induce big Cohen-Macaulay modules and classification of singularities

TL;DR

This paper explores closure operations that lead to big Cohen-Macaulay modules, characterizes their properties, and uses them to classify singularities, establishing connections with regularity and other closure types.

Contribution

It introduces the concept of Dietz closures, analyzes their properties, and demonstrates their role in classifying singularities and relating to other closure operations.

Findings

Every Dietz closure is contained in a big Cohen-Macaulay module closure.

A ring is regular if and only if all Dietz closures are trivial.

Solid, integral, and regular closures are not Dietz closures.

Abstract

Geoffrey Dietz introduced a set of axioms for a closure operation on a complete local domain R so that the existence of such a closure operation is equivalent to the existence of a big Cohen-Macaulay module. These closure operations are called Dietz closures. In complete rings of characteristic p > 0, tight closure and plus closure satisfy the axioms. In order to study these closures, we define module closures and discuss their properties. For many of these properties, there is a smallest closure operation satisfying the property. We discuss properties of big Cohen-Macaulay module closures, and prove that every Dietz closure is contained in a big Cohen-Macaulay module closure. Using this result, we show that under mild conditions, a ring R is regular if and only if all Dietz closures on R are trivial. Finally, we show that solid closure in equal characteristic 0, integral closure, and regular closure are not Dietz closures, and that all Dietz closures are contained in liftable integral closure.