Some Homological Invariants of Local PI Algebras

Abstract

During the last several years, Artin, Jorgensen, Stafford, Yekutieli, and others have generalized many homological results from the commutative case to the noncommutative connected graded case [AZ, SZ1, Jo1, Jo2, Ye]. In this paper we generalize some of those to the noncommutative local case. Our main interest is on noetherian local PI (polynomial identity) algebras, though some results hold in a more general setting. Throughout A is an algebra over a base field k, I is its Jacobson radical, and A0 = A/I. If A0 is artinian, A is called semilocal. If A0 is a simple artinian algebra (respectively, a division algebra), then A is called local (respectively, scalar local). Unless otherwise stated we are working with left modules and sometimes we use the term Aop-module for a right A-module where Aop is the opposite ring of A. A finite A-module means finitely generated over A. We use noetherian (respectively, artinian) for two-sided noetherian (respectively, two-sided artinian).