On a-invariant formulas

Let G(I) be the form ring of an ideal I of positive height in a local ring A. In this work we will provide formulas for the a-invariant of G(I). Our m ain result will only need the assumption that A is Cohen-Macaulay and that G(I) fulfills Serre's condition (S-l) where I is the analytic spread of I. As a consequence of our formula we will prove upper bounds for the reductio n exponent of I in the case that A is a regular local ring. If G(I) fulfill s Serre's condition (S-l), then this upper bound is I - 1. And in the case that G(I) is even Gorenstein, it is I - 2. (C) 2000 Academic Press.