Vector-spread monomial ideals and Eliahou-Kervaire type resolutions

TL;DR

This paper introduces vector-spread monomial ideals, generalizing t-spread ideals, and provides their minimal free resolutions, Betti numbers, and a generalized algebraic shifting theory, advancing understanding of their algebraic and combinatorial properties.

Contribution

It defines vector-spread monomial ideals, computes their resolutions and Betti numbers, and extends algebraic shifting theory to this new class.

Findings

Computed minimal free resolutions for vector-spread strongly stable ideals

Determined graded Betti numbers and Poincaré series for these ideals

Extended algebraic shifting theory to vector-spread ideals

Abstract

We introduce the class of vector-spread monomial ideals. This notion generalizes that of $t$-spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincar\'e series are determined. Finally, we consider a generalization of algebraic shifting theory for such a class of ideals.