Macaulay's theorem for vector-spread algebras

TL;DR

This paper generalizes classical theorems to ${f t}$-spread ideals, establishing a unique lex ideal with the same $f_{f t}$-vector and characterizing possible vectors, while analyzing Betti number bounds.

Contribution

It introduces the concept of ${f t}$-spread strongly stable ideals, proves the existence of a unique ${f t}$-spread lex ideal sharing the same $f_{f t}$-vector, and characterizes the possible $f_{f t}$-vectors.

Findings

Existence of a unique ${f t}$-spread lex ideal for each ${f t}$-spread strongly stable ideal.

Characterization of all possible $f_{f t}$-vectors for these ideals.

${f t}$-spread lex ideals maximize Betti numbers among ideals with the same $f_{f t}$-vector.

Abstract

Let $S=K[x_1,\dots,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let ${\bf t}=(t_1,\ldots,t_{d-1})\in{\mathbb{Z}}_{\ge 0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. To a ${\bf t}$-spread ideal $I$ in $S$, we associate a unique $f_{\bf t}$-vector and we prove that if $I$ is ${\bf t}$-spread strongly stable, then there exists a unique ${\bf t}$-spread lex ideal which shares the same $f_{\bf t}$-vector of $I$ via the combinatorics of the ${\bf t}$-spread shadows of special sets of monomials of $S$. Moreover, we characterize the possible $f_{\bf t}$-vectors of ${\bf t}$-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal-Katona. Finally, we prove that among all ${\bf t}$-spread strongly stable ideals with the same $f_{\bf t}$-vector, the ${\bf t}$-spread lex ideals have the largest Betti numbers.