Closure operations that induce big Cohen-Macaulay algebras

TL;DR

This paper explores closure operations over local domains that relate to big Cohen-Macaulay modules and algebras, introducing the Algebra Axiom to connect these closures with algebraic structures.

Contribution

It introduces the Algebra Axiom, linking Dietz closures to big Cohen-Macaulay algebras, and analyzes their properties and relationships with tight closure.

Findings

Many closure operations satisfy the Algebra Axiom.

Every Dietz closure satisfying the Algebra Axiom is contained in a big Cohen-Macaulay algebra closure.

In characteristic p > 0, such closures are contained in tight closure.

Abstract

We study closure operations over a local domain R that satisfy a set of axioms introduced by Geoffrey Dietz. The existence of a closure operation satisfying the axioms (called a Dietz closure) is equivalent to the existence of a big Cohen-Macaulay module for R. We give an additional axiom (the Algebra Axiom), 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. Several of these results rely on work from arXiv:1512.06843.