Cokernels of the Cartan Matrix and Stratifying Systems

Abstract

We study the cokernel of the application given by the Cartan Matrix $C_Λ$ of a finite dimensional $k$-algebra $Λ.$ This produces a finitely generated abelian group, the Cartan group $G_Λ,$ which is invariant under derived equivalences. We are interested in the case when $G_Λ$ 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 $G_Λ$ can be seen as a measure of how far is a standardly stratified algebra $Λ$ 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.