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On definable subcategories

Ramin Ebrahimi

Abstract

Let $\mathcal{X}$ be a skeletally small additive category. Using the canonical equivalence between two different presentations of the free abelian category over $\mathcal{X}$, we give a new and simple characterization of definable subcategories of $\rm Mod\text{-}\mathcal{X}$, and in particular definable subcategories of modules over rings. In the end, we give a conceptual proof of Auslander-Gruson-Jensen duality, which makes the duality between definable subcategories of left and right module more transparent.

On definable subcategories

Abstract

Let be a skeletally small additive category. Using the canonical equivalence between two different presentations of the free abelian category over , we give a new and simple characterization of definable subcategories of , and in particular definable subcategories of modules over rings. In the end, we give a conceptual proof of Auslander-Gruson-Jensen duality, which makes the duality between definable subcategories of left and right module more transparent.

Paper Structure

This paper contains 4 sections, 16 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Characterization of definable subcategories
  4. Auslander-Gruson-Jensen duality

Key Result

Theorem 2.3

Ade73 Let $\mathcal{X}$ be a skeletally small additive category. Then $\Romanbar{3}(\mathcal{X})$ is an abelian category. Moreover the canonical full embedding $\mathfrak{j}:\mathcal{X}\rightarrow \Romanbar{3}(\mathcal{X})$, given by $X\mapsto (0\rightarrow X\rightarrow 0)$, provides the free abelia

Theorems & Definitions (35)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • Lemma 2.8
  • Theorem 2.9
  • ...and 25 more