Homological properties of coherent semilocal rings

Abstract

Abstract.W. V. Vasconcelos and M. Auslander have studied the homological properties of coherent (noetherian) rings. In [3] some theorems for regular local rings were generalized to noetherian semilocal rings. The main aim of this paper is to discuss coherent semilocal rings. HOMOLOGICAL DIMENSIONS OF COHERENT SEMILOCAL RINGSLet R be a commutative ring with identity element, and let J be the Jacobson radical of R. The concepts and the notations that are used in this paper are consistent with those in [14].Definition. Let A be an 7<-module. A normal .4-sequence is an ordered sequence ux, u2, ... , un in J such that zz, is not a zero divisor in A and, for j > 1, each ui is not a zero divisor in A/(ux, ... , u¡_x)A. We write codR(A) -n if there exists a normal ^-sequence with n terms but no normal sequence with more than n terms. Theorem 1.1. Let R be a coherent semilocal ring such that J is finitely generated, and let A be a finitely presented R-module. Then

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