LAUSR

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 ,h 3 ,h 4 ) h = ( 1 , 3 , h 2 , h 3 , h 4 ) , where h 4 ≥2 h 4 ≥ 2 , is the h -vector of a local level K -algebra if and only if h 3 ≤3h 4 h 3 ≤ 3 h 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) 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 h 1 = h 3 and h 2 = ( h 1 + 1 2 ) , that is the h -vector is maximal. We discuss analogue problems for higher socle degrees.