Lim Cohen-Macaulay sequences of modules

TL;DR

This paper introduces lim Cohen-Macaulay sequences of modules, proves their existence in positive and mixed characteristic, and shows their implications for longstanding conjectures and the theory of big Cohen-Macaulay modules.

Contribution

It defines lim Cohen-Macaulay sequences, proves their existence in various characteristics, and links them to key conjectures and closure operations in commutative algebra.

Findings

Existence of lim Cohen-Macaulay sequences in positive characteristic.

Implication of these sequences for the positivity of Serre intersection multiplicities.

Connection to the existence of big Cohen-Macaulay modules.

Abstract

We introduce the notion of a lim Cohen-Macaulay sequence of modules. We prove the existence of such sequences in positive characteristic, and show that their existence in mixed characteristic implies the long open conjecture about positivity of Serre intersection multiplicities for all regular local rings, as well as a new proof of the existence of big Cohen-Macaulay modules. We describe how such a sequence leads to a notion of closure for submodules of finitely generated modules: this family of closure operations includes the usual notion of tight closure in characteristic $p>0$, and all of them have the property of capturing colon ideals. In fact they satisfy axioms formulated by G.~Dietz from which it follows that if a local ring $R$ has a lim Cohen-Macaulay sequence then it has a big Cohen-Macaulay module. We also prove the existence of lim Cohen-Macaulay sequences for certain rings of mixed characteristic.