A numerical characterization of the extremal Betti numbers of $t$-spread strongly stable Ideals

TL;DR

This paper provides a constructive method to characterize and build $t$-spread strongly stable ideals in polynomial rings with specified extremal Betti numbers, advancing understanding of their algebraic and combinatorial properties.

Contribution

It introduces a constructive approach to determine and realize extremal Betti numbers of $t$-spread strongly stable ideals, including existence conditions and explicit construction methods.

Findings

Characterization of extremal Betti numbers for $t$-spread strongly stable ideals.

Conditions for the existence of ideals with prescribed extremal Betti numbers.

Explicit construction method for such ideals.

Abstract

Let $K$ be a field and let $S=K[x_1,\dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is constructive. Indeed, given some positive integers $a_1,\dots,a_r$ and some pairs of positive integers $(k_1,\ell_1),\dots,(k_r,\ell_r)$, we are able to determine under which conditions there exist a $t$-spread strongly stable ideal $I$ of $S$ with $\beta_{k_i, k_i\ell_i}(I)=a_i$, $i=1, \ldots, r$, as extremal Betti numbers, and then to construct it.