Minimal Resolutions of Dominant and Semidominant Ideals

TL;DR

This paper constructs minimal resolutions for specific classes of monomial ideals, extending known classes and providing combinatorial insights into their structure and relation to generic ideals.

Contribution

It introduces minimal resolutions for dominant, 1-semidominant, and 2-semidominant ideals, expanding the understanding of their algebraic and combinatorial properties.

Findings

Dominant ideals precisely characterize when the Taylor resolution is minimal.

1-semidominant ideals are Scarf.

Minimal resolutions of 2-semidominant ideals are obtained from Taylor resolutions by face elimination.

Abstract

We construct the minimal resolutions of three classes of monomial ideals: dominant, 1-semidominant, and 2-semidominant ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections. We show that dominant ideals give a precise characterization of when the Taylor resolution is minimal, 1-semidominant ideals are Scarf, and the minimal resolutions of 2-semidominant ideals can be obtained from their Taylor resolutions by eliminating faces and facets of equal multidegree, in arbitrary order. We study the combinatorial properties of these classes of ideals and explain how they relate to generic ideals.