Homological properties of simple modules over Leavitt path algebras

TL;DR

This paper constructs explicit projective resolutions for simple modules over Leavitt path algebras associated with graphs and studies the extension spaces between these modules.

Contribution

It provides explicit constructions of projective resolutions for simple modules over Leavitt path algebras and analyzes their extension groups.

Findings

Explicit projective resolutions for simple modules over Leavitt path algebras.

Determination of the dimension of extension groups between simple modules.

Abstract

Let $K$ be any field, and let $E$ be any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra $L_K(E)$ associated to cycles and irreducible polynomials. Then we study the dimension of the $K$-vector space of the extensions between two such simple modules.