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The multigraded BGG correspondence in Macaulay2

Maya Banks, Michael K. Brown, Tara Gomes, Prashanth Sridhar, Eduardo Torres Davila, Sasha Zotine

TL;DR

The paper presents MultigradedBGG, a Macaulay2 package implementing the multigraded Bernstein–Gelfand–Gelfand correspondence between differential E-modules and S-complexes, extending the standard BGG framework to rings graded by an abelian group and to toric geometry. It introduces differential modules, free flag resolutions, and algorithms (resDM, minimizeDM, resMinFlag) for constructing and minimizing free resolutions in the differential setting, enabling robust computation of minimal structures where they exist. The multigraded BGG functors L and R are developed and applied, with practical illustrations over toric Cox rings and Hirzebruch surfaces, including finite-window computations and Tate resolutions for cohomology. A key contribution is the computation of strongly linear strands via a concrete formula and its implementation (stronglyLinearStrand), enabling explicit identification of linear substructures in multigraded free resolutions for applications in toric geometry and beyond.

Abstract

We give an overview of a Macaulay2 package for computing with the multigraded BGG correspondence. This software builds on the package BGG due to Abo-Decker-Eisenbud-Schreyer-Smith-Stillman, which concerns the standard graded BGG correspondence. In addition to implementing the multigraded BGG functors, this package includes an implementation of differential modules and their minimal free resolutions, and it contains a method for computing strongly linear strands of multigraded free resolutions.

The multigraded BGG correspondence in Macaulay2

TL;DR

The paper presents MultigradedBGG, a Macaulay2 package implementing the multigraded Bernstein–Gelfand–Gelfand correspondence between differential E-modules and S-complexes, extending the standard BGG framework to rings graded by an abelian group and to toric geometry. It introduces differential modules, free flag resolutions, and algorithms (resDM, minimizeDM, resMinFlag) for constructing and minimizing free resolutions in the differential setting, enabling robust computation of minimal structures where they exist. The multigraded BGG functors L and R are developed and applied, with practical illustrations over toric Cox rings and Hirzebruch surfaces, including finite-window computations and Tate resolutions for cohomology. A key contribution is the computation of strongly linear strands via a concrete formula and its implementation (stronglyLinearStrand), enabling explicit identification of linear substructures in multigraded free resolutions for applications in toric geometry and beyond.

Abstract

We give an overview of a Macaulay2 package for computing with the multigraded BGG correspondence. This software builds on the package BGG due to Abo-Decker-Eisenbud-Schreyer-Smith-Stillman, which concerns the standard graded BGG correspondence. In addition to implementing the multigraded BGG functors, this package includes an implementation of differential modules and their minimal free resolutions, and it contains a method for computing strongly linear strands of multigraded free resolutions.
Paper Structure (4 sections, 2 theorems, 6 equations, 2 algorithms)

This paper contains 4 sections, 2 theorems, 6 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Computing with differential modules
  3. The multigraded BGG functors
  4. Computing strongly linear strands of multigraded free resolutions

Key Result

Theorem 2.6

Let $(D, \partial_D)$ be a degree $a$ differential $R$-module. Assume its homology $H(D, \partial_D)$ is finitely generated. If either $a = 0$ or $H(D, \partial_D)$ has finite projective dimension as an $R$-module, then $(D, \partial_D)$ admits a minimal free resolution that is unique up to isomorph

Theorems & Definitions (14)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3: ABI § 2
  • Example 2.4
  • Definition 2.5
  • Theorem 2.6: BE1 Theorem 1.2
  • Example 2.8
  • Example 2.9
  • Proposition 2.10
  • Example 2.12
  • ...and 4 more