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Resolutions as directed colimits

Leonid Positselski

Abstract

A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any countably Noetherian ring). We show that any $R$-module of flat dimension $n$ is a directed colimit of countably presentable $R$-modules of flat dimension at most $n$, and any flatly coresolved $R$-module is a directed colimit of countably presentable flatly coresolved $R$-modules. If $R$ is a countably coherent ring with a dualizing complex, then any F-totally acyclic complex of flat $R$-modules is a directed colimit of F-totally acyclic complexes of countably presentable flat $R$-modules. The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorenstein-flat module over a countably coherent ring is a directed colimit of countably presentable Gorenstein-flat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions.

Resolutions as directed colimits

Abstract

A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring (e.g., any coherent ring or any countably Noetherian ring). We show that any -module of flat dimension is a directed colimit of countably presentable -modules of flat dimension at most , and any flatly coresolved -module is a directed colimit of countably presentable flatly coresolved -modules. If is a countably coherent ring with a dualizing complex, then any F-totally acyclic complex of flat -modules is a directed colimit of F-totally acyclic complexes of countably presentable flat -modules. The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorenstein-flat module over a countably coherent ring is a directed colimit of countably presentable Gorenstein-flat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions.
Paper Structure (16 sections, 70 theorems, 1 equation)

This paper contains 16 sections, 70 theorems, 1 equation.

Table of Contents

  1. Accessible Categories and Acyclic Complexes
  2. Modules of Small Presentability Rank as Small Directed Colimits
  3. Deconstructibility and Directed Colimits
  4. Two-Sided Resolutions by Accessible Classes
  5. Modules of Finite Flat Dimension
  6. Flatly Coresolved Modules
  7. Dualizing Complexes and F-Totally Acyclic Complexes
  8. F-Totally Acyclic Complexes as Directed Colimits
  9. Commutative Noetherian Rings with Countable Spectrum
  10. Gorenstein-Flat Modules as Directed Colimits
  11. Modules of Finite Injective Dimension
  12. Injectively Resolved Modules
  13. Totally Acyclic Complexes of Injectives over a Left Noetherian Ring
  14. Dualizing Complexes and Totally Acyclic Complexes of Injectives
  15. Totally Acyclic Complexes of Injectives as Directed Colimits
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $\kappa$ be a regular cardinal and $\mathsf K$ be a $\kappa$-accessible category. Let $\mathsf T\subset\mathsf K$ be a set of $\kappa$-presentable objects. Then the full subcategory $\varinjlim_{(\kappa)}\mathsf T\subset\mathsf K$ is closed under $\kappa$-directed colimits in $\mathsf K$. The ca

Theorems & Definitions (143)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Theorem 1.3
  • proof
  • Theorem 1.4
  • proof
  • Lemma 1.5
  • proof
  • ...and 133 more