Depth of symmetric algebras of certain ideals

Abstract

We compute the depth of the symmetric algebra of certain ideals in terms of the depth of the ring modulo the ideal generated by the entries of a minimal presentation matrix. The purpose of this paper is to give a formula for the depth of the symmetric algebra S(I) =EDj>0 Sj (I) for certain ideals. We also obtain an analogous result for the symmetric algebra S(I/I2) of the conormal module (as an R/I-module). Our method is a simple application of the recent results ([3], [5]) concerning the Cohen-Macaulay properties of the blow-up rings R[It] = .j>oPI and gri(R) = EDjO IP/I+l. In particular, these blow-up rings are Cohen-Macaulay for the ideals under our consideration. It is well known that the symmetric algebra need not be Cohen-Macaulay, but it turns out that these symmetric algebras have fairly large depth in any case. Let R be a noetherian local ring with infinite residue field k, and let I be an ideal. Recall that the analytic spread of I is ?(I) = dim R[It] OR k, or equivalently is the least number s of elements a1, ..., a, in I such that jk+1 = (a,, ..., as)Ik for some k > 0; the least integer k (over all such reductions) is the reduction number r of I. Note that r = O f = n _ v(I) (the minimal number of generators of I) I is generated by analytically independent elements. We say that I satisfies Gs if v(Ip) s > ht I (respectively, and ht I + J > s). Theorem 1. Let R be a local Cohen-Macaulay ring with infinite residue field, let I be an ideal with ht I > 2, analytic spread X, reduction number r, minimally generated by n = f + 1 elements, let 0 be a minimal presentation matrix of I, and assume that I satisfies Ge and ANJ-2, and that Sj(I) _ Ii and depthfR/IJ > dimR-C+r-j for1 <j <r. Then depth S(I) = depth R/II (0) + n, depth S(I/I2) = min{depth R/I1 (0) + n, dim R}. Received by the editors June 17, 1999 and, in revised form, September 1, 1999. 1991 Mathematics Subject Classification. Primary 13A30, 13H10. (?)2000 American Mathematical Society