MultigradedBGG -- Package for working with Multigraded BGG and Differential Modules

Description

This package implements the multigraded BGG correspondence, as described, for instance, in Section 2.2 of the paper "Tate resolutions on toric varieties" by Brown-Erman. Applying the BGG functor to a module over a multigraded polynomial ring gives a differential E-module, rather than a complex of E-modules; this package therefore also implements differential modules. Highlights of the package include methods for building free resolutions of differential modules; implementations of the multigraded BGG functors; and a method for computing the strongly linear strand of the minimal free resolution of a module over a multigraded polynomial ring, in the sense of the paper "Linear strands of multigraded free resolutions" by Brown-Erman.

References

Text [1] Linear strands of multigraded free resolutions (with Daniel Erman), Mathematische Annalen 390 (2024), 2707–2725 [2] Tate resolutions on toric varieties (with Daniel Erman), Journal of the European Mathematical Society, published online (2024)

Authors

Version

This documentation describes version 1.2 of MultigradedBGG.

Citation

If you have used this package in your research, please cite it as follows:

@misc{MultigradedBGGSource,
  title = {{MultigradedBGG: the multigraded BGG correspondence and differential modules. Version~1.2}},
  author = {Maya Banks and Michael K. Brown and Tara Gomes and Prashanth Sridhar and Eduardo Torres Davila and Sasha	Zotine},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- DifferentialModule -- The class of differential modules.
Functions and commands
- differential -- returns the differential of a differential module
- differentialModule -- converts a square zero matrix into a differential module
- dualRingToric -- computes the Koszul dual of a multigraded polynomial ring or exterior algebra
- foldComplex -- converts a chain complex into a differential module
- minimizeDM -- minimizes a square matrix or a differential module
- resDM -- uses a "killing cycles"-style construction to find a free resolution of a differential module
- resMinFlag -- gives a minimal free flag resolution of a differential module of degree 0
- stronglyLinearStrand -- computes the strongly linear strand of the minimal free resolution of a finitely generated graded module over a multigraded polynomial ring, provided the module is generated in a single degree.
- toricLL -- computes the BGG functor of a module over the Koszul dual exterior algebra of a multigraded polynomial ring
- toricRR -- computes the BGG functor of a module over a multigraded polynomial ring.
- unfold -- converts a differential module into a 1-periodic complex
Methods
- degree(DifferentialModule) -- returns the degree of the differential
- differential(DifferentialModule) -- see differential -- returns the differential of a differential module
- differentialModule(Matrix) -- see differentialModule -- converts a square zero matrix into a differential module
- differentialModule(Complex) -- converts a complex into a differential module
- dualRingToric(PolynomialRing) -- see dualRingToric -- computes the Koszul dual of a multigraded polynomial ring or exterior algebra
- foldComplex(Complex,ZZ) -- see foldComplex -- converts a chain complex into a differential module
- HH DifferentialModule -- computes the homology of a differential module
- image(DifferentialModule) -- computes the image of the differential of a differential module
- kernel(DifferentialModule) -- computes the kernel of the differential in a differential module.
- minimizeDM(DifferentialModule) -- see minimizeDM -- minimizes a square matrix or a differential module
- module(DifferentialModule) -- computes the underlying module of a differential module.
- resDM(DifferentialModule) -- see resDM -- uses a "killing cycles"-style construction to find a free resolution of a differential module
- resDM(DifferentialModule,ZZ) -- see resDM -- uses a "killing cycles"-style construction to find a free resolution of a differential module
- resMinFlag(DifferentialModule,ZZ) -- see resMinFlag -- gives a minimal free flag resolution of a differential module of degree 0
- ring(DifferentialModule) -- returns the ring of a differential module.
- stronglyLinearStrand(Module) -- see stronglyLinearStrand -- computes the strongly linear strand of the minimal free resolution of a finitely generated graded module over a multigraded polynomial ring, provided the module is generated in a single degree.
- toricLL(Module) -- see toricLL -- computes the BGG functor of a module over the Koszul dual exterior algebra of a multigraded polynomial ring
- toricRR(Module) -- see toricRR -- computes the BGG functor of a module over a multigraded polynomial ring.
- toricRR(Module,List) -- see toricRR -- computes the BGG functor of a module over a multigraded polynomial ring.
- unfold(DifferentialModule,ZZ,ZZ) -- see unfold -- converts a differential module into a 1-periodic complex
Symbols

For the programmer

The object MultigradedBGG is a package, defined in MultigradedBGG.m2.

The source of this document is in MultigradedBGG.m2:415:0.