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Cellular free resolutions for normalizations of toric ideals

Christine Berkesch, Lauren Cranton Heller, Gregory G. Smith, Jay Yang

TL;DR

The paper develops a combinatorial approach to free resolutions for the integral closures of toric ideals by extending Bayer–Sturmfels cellular techniques to compatible $\mathbb{Z}^n$-stratifications. Central is the ceiling stratification $\psi(p)=\lceil p\rceil$, which yields a free resolution $F_{\psi}\otimes_{S[L]} S$ of $\overline{S/I_L}$ and a generation-by-vertices description of the integral closure. This algebraic framework aligns with geometric diagonal resolutions, showing that the $\mathcal{O}_X$-complexes associated to these cellular resolutions reproduce the known resolutions of $\varphi_*\mathcal{O}_Y$ in HHL24 and BE24, with minimality criteria and direct-summand relations clarifying their relationships. The authors also extend the method to non-saturated lattices, relate it to several other diagonal-resolutions (Anderson, And, FH), and discuss broader applicability, including connections to the Favero–Huang path-algebra perspective.

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.

Cellular free resolutions for normalizations of toric ideals

TL;DR

The paper develops a combinatorial approach to free resolutions for the integral closures of toric ideals by extending Bayer–Sturmfels cellular techniques to compatible -stratifications. Central is the ceiling stratification , which yields a free resolution of and a generation-by-vertices description of the integral closure. This algebraic framework aligns with geometric diagonal resolutions, showing that the -complexes associated to these cellular resolutions reproduce the known resolutions of in HHL24 and BE24, with minimality criteria and direct-summand relations clarifying their relationships. The authors also extend the method to non-saturated lattices, relate it to several other diagonal-resolutions (Anderson, And, FH), and discuss broader applicability, including connections to the Favero–Huang path-algebra perspective.

Abstract

For any toric ideal in a polynomial ring , we provide a combinatorial description of a free resolution of the integral closure of the -module . 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.

Paper Structure

This paper contains 5 sections, 19 theorems, 56 equations, 5 figures.

Table of Contents

  1. Overview
  2. Cellular free resolutions from stratifications
  3. The ceiling stratification
  4. Applications of the ceiling stratification
  5. Applications to other resolutions of the diagonal

Key Result

Theorem 1.1

When $\psi \colon L_{\mathbb{R}} \to \mathbb{Z}^{n}$ is defined by $\psi(\bm{p}) \mathrel{\mathop :}= \left\lceil \bm{p} \right\rceil$ for all $\bm{p} \in L_{\mathbb{R}} \subseteq \mathbb{R}^{n}$, the cellular free $S$-complex $F_{\psi} \otimes_{S[L]} S$ is a $(\mathbb{Z}^n \mathbin{\!/\!} L)$-grade

Figures (5)

  • Figure 2.3: A compatible $\mathbb{Z}^4$-stratification for the closed embedding of the identity point into the second Hirzebruch surface.
  • Figure 5.4: A fundamental domain of the Anderson stratification for the diagonal embedding of the second Hirzebruch surface.
  • Figure 5.4: A fundamental domain of an $\varepsilon$-shifted stratification for the diagonal embedding of the second Hirzebruch surface.
  • Figure 5.5: A portion of the labeled cell complex giving a Favero--Huang resolution of the diagonal for $\mathbb{P}^2$. The dashed lines illustrate the underlying lattice.
  • Figure 2.11: A diagram encoding the cellular free $S$-complex arising from the closed embedding of the identity point into the second Hirzebruch surface.

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 2.1: Cell complex arising from a periodic arrangement
  • Definition 2.2
  • Remark 2.3: Cell complex arising from a toric embedding
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 43 more