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Lim Cohen-Macaulay sequences of modules

Bhargav Bhatt, Melvin Hochster, Linquan Ma

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.

Lim Cohen-Macaulay sequences of 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 , 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 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.

Paper Structure

This paper contains 24 sections, 51 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Executive summary
  3. Terminology and notation
  4. Big and small Cohen-Macaulay modules
  5. The objectives of this paper
  6. Applying small Cohen-Macaulay modules to positivity of Serre multiplicities
  7. Progress on Serre's multiplicity conjecture
  8. Alternative notions of length and rank
  9. Subadditivity
  10. $\mathcal{L}_I$ and $\mathfrak{L}_I$ for one-dimensional cases
  11. Rank
  12. Multiple Tor and Koszul homology
  13. Lim Cohen-Macaulay sequences
  14. Weakly lim Cohen-Macaulay sequences of modules
  15. Ulrich modules and (weakly) lim Ulrich sequences
  16. ...and 9 more sections

Key Result

Theorem 1

Let $(R,\,\mathfrak{m},\,K)$ be an F-finite local ring of prime characteristic $p >0$ of Krull dimension $d$, and let $M$ be any $R$-module of Krull dimension $d$. Then the sequence $\{F^n_*(M)\}_n$ is a lim Cohen-Macaulay sequence of $R$-modules

Theorems & Definitions (128)

  • Definition 1.1
  • Theorem : \ref{['nMlimcm']}
  • Theorem : \ref{['pos']}
  • proof
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 118 more