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Exact Structures and Purity

Kevin Schlegel

TL;DR

The article develops a bridge between purity in locally finitely presented categories and exact structures on the subcategory of finitely presented objects. It constructs a relative purity framework $\mathbf{P}_{\mathcal{E}}(\mathcal{A})$ via a chosen exact structure $\mathcal{E}$ on $\text{fp}\,\mathcal{A}$, and shows how $\mathcal{A}$ embeds into this purity category with the lifted structure $\bar{\mathcal{E}}$, yielding deep links between definable subcategories, Ziegler spectrum, and Serre subcategories. In the Artin-algebra case, maximal exact structures on $\text{mod}\,A$ correspond to indecomposable endofinite non-injective modules, with a module $M$ being generic exactly when $\mathcal{E}_M$ contains all almost-split sequences, and fp-idempotent ideals provide a complete description of exact structures via $\mathcal{I}_{\mathcal{E}}$ and $\mathcal{P}_{\mathcal{E}}$ together with a relative Auslander-Reiten theory. The results yield a comprehensive framework linking purity, exact structures, definable subcategories, Ziegler topology, and Auslander-Reiten theory, enabling explicit classification and computation in concrete settings such as Artin algebras. $\,$

Abstract

We relate the theory of purity of a locally finitely presented category with products to the study of exact structures on the full subcategory of finitely presented objects. Properties in the context of purity are translated to properties about exact structures. We specialize to the case of a module category over an Artin algebra and show that generic modules are in one to one correspondence with particular maximal exact structures.

Exact Structures and Purity

TL;DR

The article develops a bridge between purity in locally finitely presented categories and exact structures on the subcategory of finitely presented objects. It constructs a relative purity framework via a chosen exact structure on , and shows how embeds into this purity category with the lifted structure , yielding deep links between definable subcategories, Ziegler spectrum, and Serre subcategories. In the Artin-algebra case, maximal exact structures on correspond to indecomposable endofinite non-injective modules, with a module being generic exactly when contains all almost-split sequences, and fp-idempotent ideals provide a complete description of exact structures via and together with a relative Auslander-Reiten theory. The results yield a comprehensive framework linking purity, exact structures, definable subcategories, Ziegler topology, and Auslander-Reiten theory, enabling explicit classification and computation in concrete settings such as Artin algebras.

Abstract

We relate the theory of purity of a locally finitely presented category with products to the study of exact structures on the full subcategory of finitely presented objects. Properties in the context of purity are translated to properties about exact structures. We specialize to the case of a module category over an Artin algebra and show that generic modules are in one to one correspondence with particular maximal exact structures.
Paper Structure (11 sections, 38 theorems, 80 equations)

This paper contains 11 sections, 38 theorems, 80 equations.

Table of Contents

  1. Preliminaries
  2. Exact structures.
  3. The extension groups.
  4. Purity
  5. The embedding.
  6. The topology on the Ziegler spectrum.
  7. The correspondences.
  8. From exact structures to purity
  9. Translation of properties
  10. The case of an Artin algebra
  11. Fp-idempotent ideals

Key Result

Theorem 1

Crawley-Boevey2Let $\mathcal{A}$ be a locally finitely presented category with products. There exists a fully faithful additive functor which commutes with filtered colimits and cokernels. Moreover, ev induces an equivalence between $\mathcal{A}$ and the fp-injective objects in $\mathbf{P}(\mathcal{A})$.

Theorems & Definitions (80)

  • Theorem
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • ...and 70 more