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Mittag-Leffler modules and variants on intersection flatness

Rankeya Datta, Neil Epstein, Kevin Tucker

TL;DR

The paper develops a comprehensive framework linking intersection flatness, Ohm-Rush content, and Mittag-Leffler theory for modules over commutative rings. By introducing stabilizers and their cyclic variants, it unifies these notions with base-change, descent, and purity phenomena, proving that over Noetherian complete local rings all five notions—Mittag-Leffler, strictly Mittag-Leffler, Ohm-Rush, Ohm-Rush trace, and intersection flatness—are equivalent for flat modules. It also clarifies the behavior of content and radical content under localization, completion, and base change, and establishes local-to-global criteria that reduce global properties to their local ones, with notable consequences for Prüfer domains. The framework yields structural insights and descent techniques that enable a systematic study of singularities via Frobenius maps and test-element theory, and sets the stage for future work on $F_*R$-type algebras and related base-change phenomena. Overall, the work integrates Raynaud–Gruson’s Mittag-Leffler theory with Ohm–Rush content to produce a robust theory with broad applications to tight closure, test ideals, and Frobenius-based methods, especially in complete local contexts.

Abstract

We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.

Mittag-Leffler modules and variants on intersection flatness

TL;DR

The paper develops a comprehensive framework linking intersection flatness, Ohm-Rush content, and Mittag-Leffler theory for modules over commutative rings. By introducing stabilizers and their cyclic variants, it unifies these notions with base-change, descent, and purity phenomena, proving that over Noetherian complete local rings all five notions—Mittag-Leffler, strictly Mittag-Leffler, Ohm-Rush, Ohm-Rush trace, and intersection flatness—are equivalent for flat modules. It also clarifies the behavior of content and radical content under localization, completion, and base change, and establishes local-to-global criteria that reduce global properties to their local ones, with notable consequences for Prüfer domains. The framework yields structural insights and descent techniques that enable a systematic study of singularities via Frobenius maps and test-element theory, and sets the stage for future work on -type algebras and related base-change phenomena. Overall, the work integrates Raynaud–Gruson’s Mittag-Leffler theory with Ohm–Rush content to produce a robust theory with broad applications to tight closure, test ideals, and Frobenius-based methods, especially in complete local contexts.

Abstract

We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.
Paper Structure (19 sections, 82 theorems, 73 equations, 1 figure)

This paper contains 19 sections, 82 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and abbreviations
  4. Pure maps
  5. Expansions of radical ideals
  6. Stabilizers and cyclic stabilizers
  7. Domination and stabilizers
  8. Mittag-Leffler and strictly Mittag-Leffler modules
  9. Cyclic domination and cyclic stabilizers
  10. Content function and Ohm-Rush modules
  11. Descent of (cyclic) stabilizers
  12. Openness of pure loci
  13. Local-to-global criteria
  14. Radical content
  15. Ohm-Rush trace, intersection flatness and Mittag-Leffler modules
  16. ...and 4 more sections

Key Result

Theorem 1

rg71 Let $R$ be a commutative ring and $M$ be an $R$-module. Then $M$ is Ohm-Rush trace if and only if $M$ is a flat strictly Mittag-Leffler $R$-module.

Figures (1)

  • Figure 1: Diagram of implications for flat modules

Theorems & Definitions (183)

  • Theorem
  • Theorem : \ref{['thm:MittagORIF']}
  • Theorem : \ref{['thm:ML-SML-ORT-intersection-flat']}
  • Theorem : \ref{['thm:OR-equivalences']}
  • Theorem
  • Theorem : \ref{['thm:descentOhm-Rush']}, \ref{['cor:pure-descent-IF']}
  • Proposition : \ref{['cor:ML-open-pure-loci']}, \ref{['lem:purity-open-locus']}
  • Theorem : \ref{['thm:local-to-global-ML-OR']}, \ref{['prop:local-to-global-Prufer']}
  • Lemma 1
  • proof
  • ...and 173 more