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The homotopy theory of complete modules

Luca Pol, Jordan Williamson

TL;DR

The paper establishes that for a commutative ring $R$ and a weakly pro-regular ideal $I$, the $I$-adic completion, $L_0^I$-completion, and derived $I$-completion of complexes share the same homotopy theory via symmetric monoidal Quillen equivalences. It develops derived and Koszul approaches to completion, proves the MGM-type relationship between derived completion and $L_0^I$-completion, and shows that contramodules align with $L_0^I$-complete structures. A comprehensive model-categorical framework is built, including adic, $L_0^I$, and derived completion model structures, and the main equivalences are shown to be compatible with change of base. The results extend existing work (e.g., SWW16) to arbitrary base changes and provide precise necessary-and-sufficient conditions for when base-change preserves complete homotopy theories, thereby unifying several perspectives on completion and torsion in algebraic contexts.

Abstract

Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity, these three notions of completions interact well. We consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism $R \to S$, we then give necessary and sufficient conditions for the categories of complete $R$-complexes and the categories of complete $S$-complexes to have equivalent homotopy theories. This recovers and generalizes a result of Sather-Wagstaff and Wicklein on extended local (co)homology.

The homotopy theory of complete modules

TL;DR

The paper establishes that for a commutative ring and a weakly pro-regular ideal , the -adic completion, -completion, and derived -completion of complexes share the same homotopy theory via symmetric monoidal Quillen equivalences. It develops derived and Koszul approaches to completion, proves the MGM-type relationship between derived completion and -completion, and shows that contramodules align with -complete structures. A comprehensive model-categorical framework is built, including adic, , and derived completion model structures, and the main equivalences are shown to be compatible with change of base. The results extend existing work (e.g., SWW16) to arbitrary base changes and provide precise necessary-and-sufficient conditions for when base-change preserves complete homotopy theories, thereby unifying several perspectives on completion and torsion in algebraic contexts.

Abstract

Given a commutative ring and finitely generated ideal , one can consider the classes of -adically complete, -complete and derived -complete complexes. Under a mild assumption on the ideal called weak pro-regularity, these three notions of completions interact well. We consider the classes of -adically complete, -complete and derived -complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism , we then give necessary and sufficient conditions for the categories of complete -complexes and the categories of complete -complexes to have equivalent homotopy theories. This recovers and generalizes a result of Sather-Wagstaff and Wicklein on extended local (co)homology.

Paper Structure

This paper contains 21 sections, 38 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. I-adic completion
  3. Topologies
  4. Additional structure
  5. Homological properties
  6. L-completion
  7. Key properties
  8. Additional structure
  9. Extension to complexes
  10. The total left derived functor of I-adic completion
  11. Koszul complexes
  12. Derived completion
  13. Contramodules and L-completion
  14. Model structures
  15. Adic completion
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $R$ be a commutative ring and $I$ be a weakly pro-regular ideal. There are symmetric monoidal Quillen equivalences

Figures (1)

  • Figure 1: Completions and change of base.

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5: Yekutieli11
  • Lemma 2.6
  • proof
  • ...and 77 more