Prime divisors, asymptotic R-sequences and unmixed local rings

Abstract

A current trend in commutative algebra has been to characterize properties of prime divisors in the completion of a local ring in terms of the independence of various elements and sequences in the local ring itself. For instance, in [2] Bruns characterizes the minimal depth of prime divisors of zero in the completion R* of a local ring (R, M) as the largest number of elements which are &P-independent, for n sufficiently large. More recently, the notion of asymptotic sequence has been used to characterize the minimal depth of minimal prime divisors of zero in R*. This was done by the author in [S] and L. J. Ratliff, Jr. in [19]. (Elements x, ,..., xd form an asymptotic sequence if x,+, P U {PI PE 2*(x,,..., x,)}, where A*(Z) = Ass R/I” large n, and r” is the integral closure of I”.) Using this result as a starting point, a full-blown theory of asymptotic grade has been developed in [19] and [7]. In this paper we seek a similar characterization of the minimal depth of all prime divisors of zero in R*. In particular, we introduce the set