Some homological properties of Borel type ideals

TL;DR

This paper investigates the homological properties of Borel type ideals, including their resolutions, multiplicity, and analytic spread, and analyzes how these properties evolve for their homological shift ideals.

Contribution

It provides explicit descriptions of free resolutions, computes multiplicity and analytic spread for specific Borel ideals, and examines the behavior of their homological shift ideals.

Findings

Determined free resolutions of Borel type ideals.

Computed multiplicity and analytic spread for squarefree principal Borel ideals.

Analyzed the behavior of height, multiplicity, and analytic spread of homological shift ideals.

Abstract

We study ideals of Borel type, including $k$-Borel ideals and $t$-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behaviour of height, multiplicity and analytic spread of the homological shift ideals $HS_j(I)$ as functions of $j$, when $I$ is an equigenerated squarefree Borel ideal.