Homological behavior of idempotent subalgebras and Ext algebras

Let A be a (left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ:= (1 - e)A(1 - e) and the semi-simple right A-module S e := eA=e radA. In this paper, we investigate the relationship between the global dimensions of A and Γ, 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 Γ. We also investigate the situation where both A and Γ 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 Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture