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Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity

Leonid Positselski, Jan Stovicek

TL;DR

The paper proves a countable Govorov–Lazard-type characterization for flat quasi-coherent sheaves on quasi-compact quasi-separated schemes, showing every such sheaf is a directed colimit of locally countably presentable flat QCohs, and extends this to countably quasi-compact, countably quasi-separated schemes. It simultaneously analyzes three categories of complexes of flat QCohs, establishing directed-colimit decompositions built from locally countably presentable flats, including a countable version of the Christensen–Holm theorem for homotopy-flat complexes under semi-separated hypotheses. The second, category-theoretic part develops a general cotorsion-periodicity framework in exact categories and applies it to quasi-coherent sheaves, proving that cotorsion-periodic QCohs on quasi-compact semi-separated schemes are cotorsion, and deriving a cotorsion-based description of the derived category $\mathsf D(X\text{-qcoh}) \simeq \mathsf D(X\text{-qcoh}^{\text{cot}})$. These results yield a robust, local-to-global approach to flatness and projective-dimension properties, with broad consequences for filtrations, Ext-orthogonality, and derived-category formulations in algebraic geometry.

Abstract

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably quasi-compact, countably quasi-separated scheme. Moreover, for three categories of complexes of flat quasi-coherent sheaves, we show that all complexes in the category can be obtained as directed colimits of complexes of locally countably presentable flat quasi-coherent sheaves from the same category. In particular, on a quasi-compact semi-separated scheme, every flat quasi-coherent sheaf is a directed colimit of flat quasi-coherent sheaves of finite projective dimension. In the second part of the paper, we discuss cotorsion periodicity in category-theoretic context, generalizing an argument of Bazzoni, Cortes-Izurdiaga, and Estrada. As the main application, we deduce the assertion that any cotorsion-periodic quasi-coherent sheaf on a quasi-compact semi-separated scheme is cotorsion.

Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity

TL;DR

The paper proves a countable Govorov–Lazard-type characterization for flat quasi-coherent sheaves on quasi-compact quasi-separated schemes, showing every such sheaf is a directed colimit of locally countably presentable flat QCohs, and extends this to countably quasi-compact, countably quasi-separated schemes. It simultaneously analyzes three categories of complexes of flat QCohs, establishing directed-colimit decompositions built from locally countably presentable flats, including a countable version of the Christensen–Holm theorem for homotopy-flat complexes under semi-separated hypotheses. The second, category-theoretic part develops a general cotorsion-periodicity framework in exact categories and applies it to quasi-coherent sheaves, proving that cotorsion-periodic QCohs on quasi-compact semi-separated schemes are cotorsion, and deriving a cotorsion-based description of the derived category . These results yield a robust, local-to-global approach to flatness and projective-dimension properties, with broad consequences for filtrations, Ext-orthogonality, and derived-category formulations in algebraic geometry.

Abstract

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably quasi-compact, countably quasi-separated scheme. Moreover, for three categories of complexes of flat quasi-coherent sheaves, we show that all complexes in the category can be obtained as directed colimits of complexes of locally countably presentable flat quasi-coherent sheaves from the same category. In particular, on a quasi-compact semi-separated scheme, every flat quasi-coherent sheaf is a directed colimit of flat quasi-coherent sheaves of finite projective dimension. In the second part of the paper, we discuss cotorsion periodicity in category-theoretic context, generalizing an argument of Bazzoni, Cortes-Izurdiaga, and Estrada. As the main application, we deduce the assertion that any cotorsion-periodic quasi-coherent sheaf on a quasi-compact semi-separated scheme is cotorsion.
Paper Structure (10 sections, 42 theorems, 11 equations)

This paper contains 10 sections, 42 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Categories of Flat Modules and Complexes
  3. Flat Quasi-Coherent Sheaves as Directed Colimits I
  4. Flat Quasi-Coherent Sheaves as Directed Colimits II
  5. Complexes of Flat Quasi-Coherent Sheaves as Directed Colimits
  6. Quasi-Coherent Sheaves of Finite Projective Dimension
  7. Preliminaries on Ext-Orthogonal Classes
  8. Exact Categories with Exact Directed Colimits
  9. Cotorsion Periodicity in Category-Theoretic Context
  10. Cotorsion Periodicity for Quasi-Coherent Sheaves

Key Result

Proposition 2.1

Let $\lambda\triangleleft\kappa$ be a pair of regular cardinals, and let $\mathsf A$ be a $\lambda$-accessible category. Then $\mathsf A$ is also a $\kappa$-accessible category. The $\kappa$-presentable objects of $\mathsf A$ are precisely all the retracts of the $\kappa$-small $\lambda$-directed co

Theorems & Definitions (88)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 78 more