Cokernels of the Cartan Matrix and Stratifying Systems

TL;DR

This paper investigates the Cartan group derived from the Cartan matrix of finite dimensional algebras, revealing its invariance, finiteness in standardly stratified cases, and its role as a measure of quasi-heredity.

Contribution

It introduces the Cartan group as an invariant, proves its finiteness for standardly stratified algebras, and shows any finite abelian group can be realized as such a Cartan group.

Findings

The Cartan group is invariant under derived equivalences.

For standardly stratified algebras, the Cartan group is always finite.

Any finite abelian group can be realized as a Cartan group of some standardly stratified algebra.

Abstract

We study the cokernel of the application given by the Cartan Matrix $C_\Lambda$ of a finite dimensional $k$-algebra $\Lambda.$ This produces a finitely generated abelian group, the Cartan group $G_\Lambda,$ which is invariant under derived equivalences. We are interested in the case when $G_\Lambda$ is finite. For a standardly stratified algebra, it is shown that this group is always finite and some interesting connections with the standard modules are found. As a consequence, it is got that $G_\Lambda$ can be seen as a measure of how far is a standardly stratified algebra $\Lambda$ to be quasi-hereditary. Finally, it is also shown that any finite abelian group can be realized as the Cartan group of some standardly stratified algebra.