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 main result will only need the assumption that A is Cohen–Macaulay and that G(I) fulfills Serreʹs condition (Sl) where l is the analytic spread of I. As a consequence of our formula we will prove upper bounds for the reduction exponent of I in the case that A is a regular local ring. If G(I) fulfills Serreʹs condition (Sl), then this upper bound is l − 1. And in the case that G(I) is even Gorenstein, it is l − 2.