Exact Structures and Purity

TL;DR

This paper explores the relationship between purity in locally finitely presented categories and exact structures, especially in module categories over Artin algebras, revealing a correspondence with generic modules.

Contribution

It establishes a connection between purity properties and exact structures, and characterizes generic modules via maximal exact structures in Artin algebra modules.

Findings

Purity properties correspond to properties of exact structures.

Generic modules are in one-to-one correspondence with certain maximal exact structures.

The framework applies specifically to module categories over 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.