Homological behavior of idempotent subalgebras and Ext algebras  

作　　者：Colin Ingalls Charles Paquette 

机构地区：[1]School of Mathematics and Statistics,Carleton UniversityOttawa,ON K1S 5E6,Canada [2]Department of Mathematics and Computer Science,Royal Military College of Canada,Kingston,ON K7K 7B4,Canada

出　　处：《Science China Mathematics》2020年第2期309-320,共12页中国科学：数学（英文版）

基　　金：supported by an NSERC Discovery Grant;supported by the University of Connecticut and by the NSF CAREER grant (Grant No. DMS-1254567)

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 := e A/e rad A. 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.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 := e A/e rad A. 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.

关 键 词：global dimension Noetherian ring semiperfect ring idempotent subalgebra Cartan determinant 

分 类 号：O153[理学—数学]