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. 1. HOMOLOGICAL DIMENSIONS OF COHERENT SEMILOCAL RINGS Let 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 R-module. A normal A-sequence is an ordered sequence u1, u2, ... , un in J such that u, is not a zero divisor in A and, for i > 1, each u1 is not a zero divisor in A/(u1, ..., ul-,)A. We write codR (A) = n if there exists a normal A-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 (i) Tor-dimR =pdR(R/J): (ii) If U1 , U2, ..., Un is a normal A-sequence, then pdR(A/u(ul, ... , u,)A) = pdR(A) + n. Proof. (i) Because R/J is a finitely presented R-module and R a coherent ring, pdR(R/J) = flat-dimR(R/J) < Tor-dimR by [2, Proposition 2]. Assume that pdR(R/J) n < oo. If A is a finitely presented R-module, then Torn+,(A, R/J) 0. Hence Torn+'l(A,, RM/JM) 0, where M is any maximal ideal of R. Since R is a semilocal ring, JM = M,,. Thus TorR'f I(AM, RM /M j) = O Received by the editors July 27, 1989 and, in revised form, December 13, 1989. 1980 Mlathemnatics Slubject Classificatiori (1985 Revision). Primary 1 3DXX. co 1990 American Mathematical Society 0002-9939/90 $ 1.00 + $ 25 per page