Strictly atomic modules in definable categories

TL;DR

This paper explores strictly atomic modules within definable categories, highlighting their properties and their role as $ ext{lim}$-generators, especially over tubular algebras with irrational slopes.

Contribution

It generalizes the concept of strictly atomic modules in definable categories and examines their properties, including specific cases over tubular algebras.

Findings

Strictly atomic modules can generate definable categories without nonzero finitely presented modules.

These modules share key properties with finitely presented modules.

Application to modules over tubular algebras with irrational slopes.

Abstract

If ${\cal D}$ is a definable category then it may contain no nonzero finitely presented modules but, by a result of Makkai, there is a $\varinjlim$-generating set of strictly ${\cal D}$-atomic modules. These modules share some key properties of finitely presented modules. We consider these modules in general and then in the case that ${\cal D}$ is the category of modules of some fixed irrational slope over a tubular algebra.