Macaulay’s theorem for vector-spread algebras

Abstract

Let [Formula: see text] be the standard graded polynomial ring, with K a field, and let [Formula: see text], [Formula: see text], be a [Formula: see text]-tuple whose entries are non-negative integers. To a t-spread ideal I in S, we associate a unique [Formula: see text]-vector and we prove that if I is t-spread strongly stable, then there exists a unique t-spread lex ideal which shares the same [Formula: see text]-vector of I via the combinatorics of the t-spread shadows of special sets of monomials of S. Moreover, we characterize the possible [Formula: see text]-vectors of t-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal–Katona. Finally, we prove that among all t-spread strongly stable ideals with the same [Formula: see text]-vector, the t-spread lex ideals have the largest Betti numbers.