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

Abstract

Let $K$ be a field and let $S=K[x_1,\dots ,x_n]$ 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 ${\beta }_{k_i,k_i{\ell }_i}(I)=a_i$, $i=1,\dots ,r$, as extremal Betti numbers, and then to construct it.