Purity Relative to Classes of Finitely Presented Modules

TL;DR

This paper explores the concept of purity in modules over finite-dimensional algebras, examining differences in purity types and providing explicit descriptions for tame hereditary cases.

Contribution

It introduces new insights into purities determined by finitely presented modules and characterizes relative pure injectives for specific algebra classes.

Findings

Purities vary depending on matrix sizes.

Explicit descriptions of pure injectives for tame hereditary algebras.

Differences between left and right module purities.

Abstract

We investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different. Then we consider purities over finite-dimensional algebras, giving a general description of the relative pure injectives which we make completely explicit in the case of tame hereditary algebras.