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Hanlon-Hicks-Lazarev resolution revisited

Lev Borisov, Zengrui Han

TL;DR

This work presents a streamlined proof of the Hanlon–Hicks–Lazarev resolution for toric substacks by reducing to an affine, commutative-algebraic framework and then extending to toric stacks through localization and equivariant methods. The core technical advance is the affine HHL complex on ${\mathbb C}^n$ built from a lattice pair $(L,\psi)$, whose homology concentrates in degree zero and identifies with $\,\mathbb{C}[C\cap M]$, with higher homology vanishing. A crucial step is the decomposition into $l$-components corresponding to cellular chains on polyhedral regions in the universal cover, enabling a clean passage to the toric-stack setting. The paper then generalizes to smooth toric stacks via Thomsen–Bondal line bundles, establishing that the HHL complex resolves the pushforward of the structure sheaf of a toric subvariety and connecting to established frameworks (BE conjecture, Geraschenko–Satriano, Cox). Overall, the results provide a simpler, data-minimal route to HHL-type resolutions and clarify how these complexes behave under localization and stack quotients, with potential applications to minimal free resolutions and Betti-number formulas in toric settings.

Abstract

Hanlon, Hicks and Lazarev constructed resolutions of structure sheaves of toric substacks by certain line bundles on the ambient toric stacks. In this paper, we give a new and substantially simpler proof of their result.

Hanlon-Hicks-Lazarev resolution revisited

TL;DR

This work presents a streamlined proof of the Hanlon–Hicks–Lazarev resolution for toric substacks by reducing to an affine, commutative-algebraic framework and then extending to toric stacks through localization and equivariant methods. The core technical advance is the affine HHL complex on built from a lattice pair , whose homology concentrates in degree zero and identifies with , with higher homology vanishing. A crucial step is the decomposition into -components corresponding to cellular chains on polyhedral regions in the universal cover, enabling a clean passage to the toric-stack setting. The paper then generalizes to smooth toric stacks via Thomsen–Bondal line bundles, establishing that the HHL complex resolves the pushforward of the structure sheaf of a toric subvariety and connecting to established frameworks (BE conjecture, Geraschenko–Satriano, Cox). Overall, the results provide a simpler, data-minimal route to HHL-type resolutions and clarify how these complexes behave under localization and stack quotients, with potential applications to minimal free resolutions and Betti-number formulas in toric settings.

Abstract

Hanlon, Hicks and Lazarev constructed resolutions of structure sheaves of toric substacks by certain line bundles on the ambient toric stacks. In this paper, we give a new and substantially simpler proof of their result.

Paper Structure

This paper contains 6 sections, 7 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. HHL complexes on affine spaces
  3. Affine HHL complex
  4. Computation of homology
  5. Examples
  6. HHL complexes on smooth toric stacks

Key Result

Lemma 2.5

The degree of a monomial $\prod x_i^{a_i} e_{\sigma}$ remains invariant when applying the differential $d$. As a consequence, the complex eq-affine-HHL-complex splits into a direct sum according to the elements $l\in M$.

Figures (2)

  • Figure 1: The Bondal stratification on the universal cover and the torus
  • Figure 2: The stratification on $[0,1]^2$

Theorems & Definitions (24)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.7: Farkas' Lemma
  • Lemma 2.8
  • ...and 14 more