Cellular free resolutions for normalizations of toric ideals

TL;DR

This paper develops a combinatorial approach to construct free resolutions of the integral closure of toric ideals, extending Bayer--Sturmfels' theory, and applies it to unify and compare resolutions of toric varieties.

Contribution

It introduces new cellular complexes for free resolutions of normalized toric ideals, extending existing theories and unifying various constructions.

Findings

Provides a combinatorial description of resolutions for normalized toric ideals.

Unifies different constructions for resolutions of toric embeddings.

Offers a framework to compare locally free resolutions of toric subvarieties.

Abstract

For any toric ideal $I$ in a polynomial ring $S$, we provide a combinatorial description of a free resolution of the integral closure of the $S$-module $S/I$. These new complexes arise from an extension of Bayer--Sturmfels' theory of cellular free resolutions. As applications, we unify several constructions for a resolution of the diagonal embedding of a toric variety, and compare the locally free resolutions for toric subvarieties introduced by Hanlon--Hicks--Lazarev and Brown--Erman.