Artinian level algebras of socle degree 4

摘要

In this paper we study the O-sequences of local (or graded) K-algebras of socle degree 4. More precisely, we prove that an O-sequence h = (1,3, h(2), ha, h(4)), where h(4) = 2, is the h-vector of a local level K-algebra if and only if h(3) = 3h(4). A characterization is also presented for Gorenstein O-sequences. In each of these cases we give an effective method to construct a local level K-algebra with a given h-vector. Moreover we refine a result of Elias and Rossi by showing that if h = (1, h(1), h(2), h(3), 1) is a unimodal Gorenstein O-sequence, then h forces the corresponding Gorenstein K-algebra to be canonically graded if and only if h(1) = h(3) and h(2) = ((2) (h1+1)), that is the h-vector is maximal. We discuss analogue problems for higher socle degrees. (C) 2018 Elsevier Inc. All rights reserved.