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

Abstract

Let K be a field and let \(S=K[x_1,\ldots ,x_n]\) be a standard polynomial ring over a field K. We characterize the extremal Betti numbers, values as well as 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 \((k_1,\ell _1),\ldots ,(k_r,\ell _r)\), we are able to determine under which conditions there exists 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.