Cokernels of the Cartan Matrix and Stratifying Systems

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.