N ov 2 00 4 When is there a unique socle - vector associated to a given h - vector ? FABRIZIO

First we construct an interesting bijection between the set of h-vectors and the set of socle-vectors of artinian algebras. As a simple consequence, we find the minimum codimension that an artinian algebra with a given socle-vector can have. Then we address the main problem of this paper: determining when there is a unique socle-vector associated to a given h-vector. In Sections 3 and 5 we work in arbitrary codi-mension r, first giving some conditions, based on homological results, that guarantee the uniqueness of the socle-vector, and then, by Inverse Systems, studying some important cases where the uniqueness fails. In Section 4, we focus our attention on codimension r ≤ 3, where we are able to solve completely the above problem, i.e. we characterize the h-vectors that admit a unique socle-vector.