Cohen-Macaulay Rees algebras and their specialization

Abstract

The Rees algebra of an ideal J in a commutative ring R is by definition the graded algebra. where t is an indeterminate. In this paper we are concerned with proving that certain Rees algebras are Cohen-Macaulay, and with answering, under strong hypotheses, the question: if R = S/N is a factor-ring of S and I = JR, when is .$(I, R) a specialization of ..$(J. S); that is, when is the natural epimorphism R @ .A(J, S) + .R(I, R) an isomorphism? These questions have interest partly because if S and .S’(J, S) are Cohen-Macaulay, then so is gr, S := S/J@ J/J’... [ 16 1 and under these hypotheses if N is perfect and R @ .S(J, S) = .$(I, R), then R and .7(1. R) are Cohen-Macaulay too. Thus, gr,R is Cohen-Macaulay, and its torsion freeness and normality, for exmple, can be characterized in terms of analytic spreads, as in [ 161; see Section 3 for details. We deal with the specialization question in Section 1. It is answered for the case where, as above, S and .2(J, S) are Cohen-Macaulay and N is 202