SYMBOLIC POWERS, SERRE CONDITIONS AND COHEN-MACAULAY REES-ALGEBRAS

Let R be a Cohen-Macaulay ring and I an unmixed ideal of height g whic h is generically a complete intersection and satisfies I(n) = I(n) for all n greater-than-or-equal-to 1. Under what conditions will the Rees algebra be Cohen-Macaulay or have good depth? A series of partial ans wers to this question is given, relating the Serre condition (S(r)) of the associated graded ring to the depth of the Rees algebra. A useful device in arguments of this nature is the canonical module of the Ree s algebra. By making use of the technique of the fundamental divisor, it is shown that the canonical module has the expected form: omega(R)[ It] congruent-to (t(1 - t)g-2).