Homological behavior of idempotent subalgebras and Ext algebras

Abstract

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e~:~=~eA/e\,{\rm~rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):=\Ext^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is Artinian of finite global dimension, then $A$ has finite global dimension if and only if so does $\Gamma$. We also investigate the situation where both $A$ and $\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.