Closure Operations that Induce Big Cohen-Macaulay Modules and Algebras, and Classification of Singularities.

Abstract

Geoffrey Dietz introduced a set of axioms for a closure operation on a complete local domain such that the existence of a closure operation satisfying these axioms is equivalent to the existence of a big Cohen-Macaulay module. These are called Dietz closures. In 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. We also 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. We give an additional axiom for a closure operation such that the existence of a Dietz closure satisfying this axiom is equivalent to the existence of a big Cohen-Macaulay algebra. We prove that many closure operations satisfy the Algebra Axiom, whether or not they are Dietz closures. We discuss the smallest big Cohen-Macaulay algebra closure on a given ring, and show that every Dietz closure satisfying the Algebra Axiom is contained in a big Cohen-Macaulay algebra closure. This leads to proofs that in rings of characteristic p > 0, every Dietz closure satisfying the Algebra Axiom is contained in tight closure, and there exist Dietz closures that do not satisfy the Algebra Axiom.