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Balanced systems for $\mathrm{Hom}$

Víctor Becerril, Octavio Mendoza, Marco A. Pérez

Abstract

From the notion of (co)generator in relative homological algebra, we present the concept of finite balanced system $[(\mathcal{X} , ω); (ν, \mathcal{Y})]$ as a tool to induce balanced pairs $(\mathcal{X} , \mathcal{Y} )$ for the $\mathrm{Hom}$ functor with domain determined by the finiteness of homological dimensions relative to $\mathcal{X}$ and $\mathcal{Y}$. This approach to balance will cover several well known ambients where right derived functors of $\mathrm{Hom}$ are obtained relative to certain classes of objects in an abelian category, such as Gorenstein projective and injective modules and chain complexes, Gorenstein modules relative to Auslander and Bass classes, among others.

Balanced systems for $\mathrm{Hom}$

Abstract

From the notion of (co)generator in relative homological algebra, we present the concept of finite balanced system as a tool to induce balanced pairs for the functor with domain determined by the finiteness of homological dimensions relative to and . This approach to balance will cover several well known ambients where right derived functors of are obtained relative to certain classes of objects in an abelian category, such as Gorenstein projective and injective modules and chain complexes, Gorenstein modules relative to Auslander and Bass classes, among others.
Paper Structure (4 sections, 21 theorems, 76 equations)

This paper contains 4 sections, 21 theorems, 76 equations.

Table of Contents

  1. Preliminaries
  2. Relative extension bifunctors
  3. Balance systems and induced balanced pairs
  4. Applications and examples

Key Result

Lemma 2.1

The complex $M_{\bullet \geq -1}$ is ${\rm Hom}_{\mathcal{A}}(\mathcal{X},-)$-acyclic if, and only if, $\eta_i$ is ${\rm Hom}_{\mathcal{A}}(\mathcal{X},-)$-acyclic for every $i \in \mathbb{N}$. Moreover, if any of these conditions holds true and for every $j \in \mathbb{N} \cup \{ -1\}$ there is an

Theorems & Definitions (45)

  • Lemma 2.1: acyclicity condition for Hom
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: relative horseshoe lemma
  • Lemma 2.5: uniqueness up to homotopy
  • Definition 2.6
  • Proposition 2.7: properties of relative extension bifunctors
  • proof
  • Definition 3.1
  • Proposition 3.2: relative derived functors
  • ...and 35 more