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ResolutionsOfStanleyReisnerRings :: ResolutionsOfStanleyReisnerRings

ResolutionsOfStanleyReisnerRings -- Resolutions of Stanley-Reisner rings over certain parameter rings.

Description

ResolutionsOfStanleyReisnerRings is a package for computing certain systems of parameters and finding the Betti numbers of the resolution of the Stanley-Reisner ring of a simplicial complex over certain parameter rings.

Given a simplicial complex D, there is a universal system of parameters (USOP) [AR,GS,HM,S] where each parameter i is a sum of the faces of D of dimension i+1. The colorful system of parameters (KSOP) is obtained by giving a proper d-vertex coloring to a balanced simplicial complex B of dimension d-1, and then summing over the colors so that each parameter j is the sum of vertices of color j. This package includes routines for computing both USOP and KSOP for simplicial complexes and for computing the graded Betti numbers of the resolutions of the Stanley-Reisner rings over the graded parameter rings.

References:

[AR] A. Adams and V. Reiner, A Colorful Hochster Formula and Universal Parameters for Face Rings. Preprint, 2020; arXiv:2007.13021.

[GS] A.M. Garsia and D. Stanton, Group actions of Stanley-Reisner rings and invariants of permutation groups. Adv. in Math. 51 (1984), 107–201.

[HM] J. Herzog and S. Moradi, Systems of parameters and the Cohen–Macaulay property. Preprint, 2020; arXiv:2006.16549.

[S] D.E. Smith, On the Cohen-Macaulay property in commutative algebra and simplicial topology. Pac. J. Math. 141 (1990), 165–196.

The goal of this work was primarily to help compute examples to provide evidence for Conjecture 6.1 [AR]. I have tried to generalize most of the functionality to make it useful in other areas. This is work in progress and many interesting pieces are still missing. All suggestions and contributions are welcome.

Author

Version

This documentation describes version 0.1 of ResolutionsOfStanleyReisnerRings.

Source code

The source code from which this documentation is derived is in the file ResolutionsOfStanleyReisnerRings.m2.

Exports

  • Functions and commands
    • barycentricSubdivision -- the barycentric subdivision of a simplicial complex.
    • colorfulPresentation -- A presentation of the Stanley-Reisner ring of the barycentric subdivision of a simplicial complex over its colorful parameter ring
    • colorfulSOP -- Colorful System of Parameters (KSOP)
    • equalBettiTally -- Computes the Betti table of two modules and compares for equality
    • sumBetti -- Computes the sum of Betti numbers.
    • universalPresentation -- A presentation of the Stanley-Reisner ring over its universal parameter ring
    • universalSOP -- Universal System of Parameters (USOP)
  • Methods
    • "barycentricSubdivision(List)" -- see barycentricSubdivision -- the barycentric subdivision of a simplicial complex.
    • "barycentricSubdivision(SimplicialComplex)" -- see barycentricSubdivision -- the barycentric subdivision of a simplicial complex.
    • "colorfulPresentation(List)" -- see colorfulPresentation -- A presentation of the Stanley-Reisner ring of the barycentric subdivision of a simplicial complex over its colorful parameter ring
    • "colorfulPresentation(SimplicialComplex)" -- see colorfulPresentation -- A presentation of the Stanley-Reisner ring of the barycentric subdivision of a simplicial complex over its colorful parameter ring
    • "colorfulSOP(SimplicialComplex)" -- see colorfulSOP -- Colorful System of Parameters (KSOP)
    • "colorfulSOP(SimplicialComplex,ZZ)" -- see colorfulSOP -- Colorful System of Parameters (KSOP)
    • "equalBettiTally(Module,Module)" -- see equalBettiTally -- Computes the Betti table of two modules and compares for equality
    • "sumBetti(Module)" -- see sumBetti -- Computes the sum of Betti numbers.
    • "universalPresentation(List)" -- see universalPresentation -- A presentation of the Stanley-Reisner ring over its universal parameter ring
    • "universalPresentation(SimplicialComplex)" -- see universalPresentation -- A presentation of the Stanley-Reisner ring over its universal parameter ring
    • "universalSOP(List)" -- see universalSOP -- Universal System of Parameters (USOP)
    • "universalSOP(SimplicialComplex)" -- see universalSOP -- Universal System of Parameters (USOP)

For the programmer

The object ResolutionsOfStanleyReisnerRings is a package.