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Flatness in finitely accessible additive categories

Manuel Cortés-Izurdiaga

TL;DR

This work extends the theory of flatness from module categories to finitely accessible additive categories by leveraging Crawley–Boevey's representation theorem, which identifies such categories with flat functors in the functor category $({\\rm fp}(\\mathcal{A})^{\\mathrm{op}}, \\mathrm{Ab})$. It develops a thorough framework based on the torsion theory $\\tau=(\\mathcal{T}_{\\tau},\\mathcal{F}_{\\tau})$ cogenerated by the flat objects $\\mathcal{S}\\textrm{-}\\mathrm{Flat}$, with purity playing a central role in characterizing flatness and in describing morphisms and epis/monos in the flat setting. The paper provides complete criteria for when $\\mathcal{S}\\textrm{-}\\mathrm{Flat}$ is preabelian or abelian, connects these properties to the local coherence of $\\mathrm{Mod}-\\mathcal{S}$ and to the weak dimension, and proves that, under suitable hypotheses, every flat object is a direct union of small flat subobjects. It also identifies conditions for the existence of enough flat and/or projective objects and translates these conditions into the functor-category/torsion-theory language, with concrete consequences in the ring case. Overall, the results offer a robust, transferable toolkit for analyzing flat objects and their homological behavior across finitely accessible additive categories and their ring-theoretic specializations, impacting how purity, cotorsion theories, and deconstructibility interact in this broader setting.

Abstract

Motivated by some problems proposed by Cuadra and Simson related to flat objects in finitely accessible Grothendieck categories, we study flatness in the more general setting of finitely accessible additive categories. For such category $\mathcal{A}$, we characterize when $\mathcal{A}$ is preabelian and abelian. We prove that if the class of flat objects in $\mathcal A$ is closed under pure subobjects, then every flat object is a direct union of \textit{small} flat subobjects. Finally, we characterize when $\mathcal{A}$ has enough flat and projective objects and we prove that, in this case, the class of flat objects is closed under pure subobjects.

Flatness in finitely accessible additive categories

TL;DR

This work extends the theory of flatness from module categories to finitely accessible additive categories by leveraging Crawley–Boevey's representation theorem, which identifies such categories with flat functors in the functor category . It develops a thorough framework based on the torsion theory cogenerated by the flat objects , with purity playing a central role in characterizing flatness and in describing morphisms and epis/monos in the flat setting. The paper provides complete criteria for when is preabelian or abelian, connects these properties to the local coherence of and to the weak dimension, and proves that, under suitable hypotheses, every flat object is a direct union of small flat subobjects. It also identifies conditions for the existence of enough flat and/or projective objects and translates these conditions into the functor-category/torsion-theory language, with concrete consequences in the ring case. Overall, the results offer a robust, transferable toolkit for analyzing flat objects and their homological behavior across finitely accessible additive categories and their ring-theoretic specializations, impacting how purity, cotorsion theories, and deconstructibility interact in this broader setting.

Abstract

Motivated by some problems proposed by Cuadra and Simson related to flat objects in finitely accessible Grothendieck categories, we study flatness in the more general setting of finitely accessible additive categories. For such category , we characterize when is preabelian and abelian. We prove that if the class of flat objects in is closed under pure subobjects, then every flat object is a direct union of \textit{small} flat subobjects. Finally, we characterize when has enough flat and projective objects and we prove that, in this case, the class of flat objects is closed under pure subobjects.
Paper Structure (6 sections, 45 theorems, 17 equations)

This paper contains 6 sections, 45 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Functor categories
  3. Torsion theories in functor categories
  4. When is ${\mathcal{S}}\textrm{-}\mathrm{Flat}$ abelian?
  5. Flat and projective objects in ${\mathcal{S}}\textrm{-}\mathrm{Flat}$
  6. The particular case of the ring

Key Result

Theorem A

Let $\mathcal{A}$ be a finitely accessible additive category. Then, the following are equivalent: Moreover, $\mathcal{A}$ is abelian if and only if $({\rm fp}(\mathcal{A}),\mathop{\mathrm{Ab}}\nolimits)$ is locally coherent and for every object $M$ in $\mathcal{A}$ and subobject $K$ of $M$ in $({\rm fp}(\mathcal{A})^{\mathop{\mathrm{op}}\nolimits},\mathop{\mathrm{Ab}}\nolimits)$, the inclusion $

Theorems & Definitions (88)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • ...and 78 more