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Relative tilting theory in abelian categories I: Auslander-Buchweitz-Reiten approximations theory in subcategories and cotorsion pairs

Alejandro Argudín Monroy, Octavio Mendoza Hernández

Abstract

In this paper we introduce a special kind of relative (co)resolutions associated to a pair of classes of objects in an abelian category $\mathcal{C}.$ We will see that, by studying these relative (co)resolutions, we get a possible generalization of a part of the Auslander-Buchweitz approximation theory that is useful for developing $n$-$\mathcal{X}$-tilting theory in [4]. With this goal, new concepts as $\mathcal{X}$-complete and $\mathcal{X}$-hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category [5]. Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of $n$-$\mathcal{X}$-tilting theory in [4].

Relative tilting theory in abelian categories I: Auslander-Buchweitz-Reiten approximations theory in subcategories and cotorsion pairs

Abstract

In this paper we introduce a special kind of relative (co)resolutions associated to a pair of classes of objects in an abelian category We will see that, by studying these relative (co)resolutions, we get a possible generalization of a part of the Auslander-Buchweitz approximation theory that is useful for developing --tilting theory in [4]. With this goal, new concepts as -complete and -hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category [5]. Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of --tilting theory in [4].

Paper Structure

This paper contains 10 sections, 35 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Relative homological dimensions
  5. Relative $n$-quotients and $n$-subobjects
  6. Cotorsion-like pairs and related notions
  7. Relative hereditary cotorsion-like pairs
  8. Relative complete pairs
  9. Relative resolutions, coresolutions and related classes
  10. The class $\Gennr[M][n][\mathcal{X}]$

Key Result

Theorem 2.1

mitchell Let $0\to N\to M\to K\to 0$ be an exact sequence in the abelian category $\mathcal{C}$. Then, for any $X\in\mathcal{C},$ there is a long exact sequence of abelian groups induced by $\mathrm{Hom}_\mathcal{C}(X,-)$

Theorems & Definitions (75)

  • Theorem 2.1
  • Lemma 2.2: Shifting Lemma
  • proof
  • Theorem 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Corollary 2.8
  • ...and 65 more