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,h2,h3,h4), where h4≥2, is the h-vector of a local level K-algebra if and only if h3≤3h4. 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,h1,h2,h3,1) is a unimodal Gorenstein O-sequence, then h forces the corresponding Gorenstein K-algebra to be canonically graded if and only if h1=h3and h2=(h1+12), that is the h-vector is maximal. We discuss analogue problems for higher socle degrees.
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