Homological behavior of idempotent subalgebras and Ext algebras

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2020,（2）
• 页码范围: 309-320
• 页数: 12页
• 语种: 英文

摘要

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 Se:=eA/eradA.In this paper,we investigate the relationship between the global dimensions of A and Γ,by using the homological properties of Se.More precisely,we consider the Yoneda ring Y(e):=Ext*A(Se,Se) 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.