Analytic spread modulo an element and symbolic rees algebras

Abstract

Let (T, N) be a d-dimensional regular local ring, P a prime ideal of T having height equal d 1, and f~ P. Within this setting we are interested in computing the analytic spread of P/f T. We have two reasons for seeking this analytic spread. The first is to identify prime ideals in local hypersurface rings whose analytic spreads and heights are the same. The second is to aid in a better understanding of the symbolic Rees algebra, @ P’“‘t”, of P (the ideal Pen) is the nth symbolic power of P). Our interest in hypersurface rings essentially originated in [HUG]. There we were interested in determining when powers and symbolic powers are equal for a prime ideal in a local hypersurface ring. For a prime ideal q adjacent to the maximal ideal (i.e., height (M/q) = l), the only time such an equality can occur for all large powers is when the analytic spread of q equals the height of q. Equality does not, however, necessarily occur under this analytic spread condition (cf. [Hu2, Example 2. lo]). In order to force equality for all powers, one must make additional assumptions on the associated graded ring of P/f T= q (cf. [Hu2, Theorem 2.11). Section 2 of the present article contains some general methods for recognizing the analytic spread of P/f T, and Section 3 contains some illustrative examples where the analytic spread of P/f T equals the height of P/f T, The Noetherian property of the symbolic Rees algebra has created recent interest due to a question of Cowsik. In connection with his result in [C] that @ P’“‘t” Noetherian implies P is a set-theoretic complete intersection when T is a regular local ring and dim( T/P) = 1, Cowsik asked whether @ P(“)t” is Noetherian for every prime ideal in a regular local ring. An example due to P. Roberts [Ro] showed the answer is no in general, but the question has continued to draw attention [Hl, H2, KR, 0, E]. The