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Homological theory of representations having pure acyclic injective resolutions

Gang Yang, Qihui Li, Junpeng Wang

Abstract

Let $Q$ be a quiver and $R$ an associative ring. A representation by $R$-modules of $Q$ is called strongly fp-injective if it admits a pure acyclic injective resolution in the category of representations. It is shown that such representations possess many nice properties. We characterize strongly fp-injective representations under some mild assumptions, which is closely related to strongly fp-injective $R$-modules. Subsequently, we use such representations to define relative Gorenstein injective representations, called Gorenstein strongly fp-injective representations, and give an explicit characterization of the Gorenstein strongly fp-injective representations of right rooted quivers. As an application, a model structure in the category of representations is given.

Homological theory of representations having pure acyclic injective resolutions

Abstract

Let be a quiver and an associative ring. A representation by -modules of is called strongly fp-injective if it admits a pure acyclic injective resolution in the category of representations. It is shown that such representations possess many nice properties. We characterize strongly fp-injective representations under some mild assumptions, which is closely related to strongly fp-injective -modules. Subsequently, we use such representations to define relative Gorenstein injective representations, called Gorenstein strongly fp-injective representations, and give an explicit characterization of the Gorenstein strongly fp-injective representations of right rooted quivers. As an application, a model structure in the category of representations is given.
Paper Structure (4 sections, 24 theorems, 33 equations)

This paper contains 4 sections, 24 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Strongly fp-injective representations
  4. Relative Gorenstein injective objects

Key Result

Proposition 2.1

Let $Q$ be a quiver, and $\{W_{\alpha}\}_{\alpha\text{ }\mathrm{ ordinal}}$ the transfinite sequence, of subsets of $Q_0$, defined as the above. Then the following statements hold:

Theorems & Definitions (45)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Definition 3.1
  • Proposition 3.3
  • proof
  • ...and 35 more