This is ResolutionsOfStanleyReisnerRings.info, produced by Macaulay2, version 1.25.05

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
======

  * Ashleigh Adams (see http://www.ashleigh-adams.com ) <adams869@umn.edu>

Version
=======

This documentation describes version 0.1 of ResolutionsOfStanleyReisnerRings.

Citation
========

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

+-----------------------------------------------------------------------------+
|@misc{ResolutionsOfStanleyReisnerRingsSource,                                |
|  title = {{ResolutionsOfStanleyReisnerRings: A \emph{Macaulay2} package.    |
|  author = {Ashleigh Adams},                                                 |
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages   |
|}                                                                            |
|-----------------------------------------------------------------------------|
|Version~0.1}},                                                               |
|                                                                             |
|                                                                             |
|}}                                                                           |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * 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 (Macaulay2Doc),
defined in ResolutionsOfStanleyReisnerRings.m2.

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:296:0.

barycentricSubdivision -- the barycentric subdivision of a simplicial complex.
******************************************************************************

  * Usage:
        barycentricSubdivision D
        barycentricSubdivision F
  * Inputs:
      * D, an abstract simplicial complex (SimplicialComplexes),
      * F, a list (Macaulay2Doc), a list of facets defining a
        simplicial complex
  * Outputs:
      * an abstract simplicial complex (SimplicialComplexes),
        the barycentric subdivision of
        a simplicial complex of D

Description
===========

The following example uses the simplicial complex on four vertices composed of
two disconnected edges.

+------------------------------------------------------------------+
|i1 : S = QQ[a,b,c,d];                                             |
+------------------------------------------------------------------+
|i2 : F = {a*b,c*d};                                               |
+------------------------------------------------------------------+
|i3 : D = simplicialComplex F                                      |
|                                                                  |
|o3 = simplicialComplex | cd ab |                                  |
|                                                                  |
|o3 : SimplicialComplex                                            |
+------------------------------------------------------------------+
|i4 : barycentricSubdivision D                                     |
|                                                                  |
|o4 = simplicialComplex | v_0v_4v_6 v_0v_3v_6 v_0v_2v_5 v_0v_1v_5 ||
|                                                                  |
|o4 : SimplicialComplex                                            |
+------------------------------------------------------------------+
|i5 : barycentricSubdivision F                                     |
|                                                                  |
|o5 = simplicialComplex | v_0v_4v_6 v_0v_3v_6 v_0v_2v_5 v_0v_1v_5 ||
|                                                                  |
|o5 : SimplicialComplex                                            |
+------------------------------------------------------------------+

See also
========

  * SimplicialComplexes -- exploring abstract
    simplicial complexes within commutative algebra
  * Posets -- a package for working with partially ordered
    sets
  * SimplicialDecomposability -- various
    decomposability routines for simplicial complexes.

Ways to use barycentricSubdivision:
===================================

  * "barycentricSubdivision(List)"
  * "barycentricSubdivision(SimplicialComplex)"
  * barycentricSubdivision(SimplicialComplex,Ring) (SimplicialComplexes),
    -- create the barycentric subdivision of a simplicial complex
  * barycentricSubdivision(SimplicialMap,Ring,Ring) (SimplicialComplexes)
    -- create the map between barycentric subdivisions corresponding to a
    simplicial map

For the programmer
==================

The object barycentricSubdivision (SimplicialComplexes),
is a
method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:329:0.

colorfulPresentation -- A presentation of the Stanley-Reisner ring of the barycentric subdivision of a simplicial complex over its colorful parameter ring
**********************************************************************************************************************************************************

  * Usage:
        colorfulPresentation D
        colorfulPresentation F
  * Inputs:
      * D, an abstract simplicial complex (SimplicialComplexes),
      * F, a list (Macaulay2Doc), A list of facets defining a
        simplicial complex D
  * Outputs:
      * a module (Macaulay2Doc), A graded presentation of the
        Stanley-Reisner ring over its universal parameter ring

Description
===========

Since the barycentric subdivision of a simplicial complex D is a balanced
simplicial complex B, i.e., there exists a dim(D)+1 proper vertex coloring,
this colorfulPresentation takes in the barycentric subdivision of a simplicial
complex, computes a colorful system of parameters for the Stanley-Reisner ring
of B and then returns this quotient ring as an N-graded module over the
colorful parameter ring.

+--------------------------------------------------+
|i1 : S = QQ[a..e];                                |
+--------------------------------------------------+
|i2 : F = {a*b*c,c*d,e}                            |
|                                                  |
|o2 = {a*b*c, c*d, e}                              |
|                                                  |
|o2 : List                                         |
+--------------------------------------------------+
|i3 : D = simplicialComplex F                      |
|                                                  |
|o3 = simplicialComplex | e cd abc |               |
|                                                  |
|o3 : SimplicialComplex                            |
+--------------------------------------------------+
|i4 : colorfulPresentation D                       |
|                                                  |
|o4 = cokernel {0} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {3} | 0 0 0 0 |                     |
|              {1} | d 0 0 0 |                     |
|              {1} | 0 d 0 c |                     |
|              {2} | 0 0 0 0 |                     |
|              {2} | 0 0 0 0 |                     |
|              {2} | 0 0 d 0 |                     |
|                                                  |
|                                            9     |
|o4 : QQ[b..d]-module, quotient of (QQ[b..d])      |
+--------------------------------------------------+
|i5 : M = colorfulPresentation F                   |
|                                                  |
|o5 = cokernel {0} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {3} | 0 0 0 0 |                     |
|              {1} | d 0 0 0 |                     |
|              {1} | 0 d 0 c |                     |
|              {2} | 0 0 0 0 |                     |
|              {2} | 0 0 0 0 |                     |
|              {2} | 0 0 d 0 |                     |
|                                                  |
|                                            9     |
|o5 : QQ[b..d]-module, quotient of (QQ[b..d])      |
+--------------------------------------------------+
|i6 : degrees M                                    |
|                                                  |
|o6 = {{0}, {1}, {1}, {3}, {1}, {1}, {2}, {2}, {2}}|
|                                                  |
|o6 : List                                         |
+--------------------------------------------------+

See also
========

  * Module (Macaulay2Doc) -- the class of all modules
  * colorfulSOP -- Colorful System of Parameters (KSOP)

Ways to use colorfulPresentation:
=================================

  * "colorfulPresentation(List)"
  * "colorfulPresentation(SimplicialComplex)"

For the programmer
==================

The object colorfulPresentation, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:519:0.

colorfulSOP -- Colorful System of Parameters (KSOP)
***************************************************

  * Usage:
        colorfulSOP(B,n)
        colorfulSOP(D)
  * Inputs:
      * B, an abstract simplicial complex (SimplicialComplexes),
        the barycentric subdivision of
        a simplicial complex D
      * n, an integer (Macaulay2Doc), the number of vertices of D
      * D, an abstract simplicial complex (SimplicialComplexes),
        any simplicial complex for
        which you want the KSOP for its barycentric subdivision
  * Outputs:
      * a list (Macaulay2Doc), a list of sums of ring elements of
        the polynomial ring of B

Description
===========

The barycentric subdivision of a simplicial complex is a balanced simplicial
complex, and so in this version we require the input of the barycentric
subdivision of a simplicial complex.

The examples use the simplicial complex D on five vertices consisting of an
isolated vertex and triangle of dimension 2 attached to an edge. The first
example uses the barycentric subdivision of D. When considering the barycentric
subdivision of a simplicial complex it is often useful to remove the minimal
element of its face poset. In the next example, we demonstrate how this can be
accomplished.

+-------------------------------------------------------+
|i1 : S = QQ[a..e];                                     |
+-------------------------------------------------------+
|i2 : D = simplicialComplex {a*b*c,c*d,e};              |
+-------------------------------------------------------+
|i3 : n = # vertices D                                  |
|                                                       |
|o3 = 5                                                 |
+-------------------------------------------------------+
|i4 : ComplexIncludingMinElt = barycentricSubdivision D;|
+-------------------------------------------------------+
|i5 : R = ring(ComplexIncludingMinElt);                 |
+-------------------------------------------------------+
|i6 : B = faceDelete(R_0,ComplexIncludingMinElt);       |
+-------------------------------------------------------+
|i7 : colorfulSOP(B,n)                                  |
|                                                       |
|o7 = {v  + v  + v  + v  + v , v  + v  + v  + v , v  }  |
|       1    2    3    4    5   6    7    8    9   10   |
|                                                       |
|o7 : List                                              |
+-------------------------------------------------------+

Also included is the option to input any simplicial complex D in order to
obtain the colorful system of parameters for the face ring of the corresponding
barycentric subdivision of D.

+------------------------------------------------------+
|i8 : S = QQ[a..e];                                    |
+------------------------------------------------------+
|i9 : D = simplicialComplex {a*b*c,c*d,e};             |
+------------------------------------------------------+
|i10 : colorfulSOP(D)                                  |
|                                                      |
|o10 = {v  + v  + v  + v  + v , v  + v  + v  + v , v  }|
|        1    2    3    4    5   6    7    8    9   10 |
|                                                      |
|o10 : List                                            |
+------------------------------------------------------+

Caveat
======

This current version requires the input of the barycentric subdivision of a
simplicial complex D without the minimal element of the face poset, which
corresponds to the empty set of D. This must be accomplished by using the
barycentricSubdivision method, included in this package. It also has the option
of inputting any simplicial complex. The system then takes the barycentric
subdivision of D and removes its minimal element in order to obtain the
colorful system of parameters.

See also
========

  * barycentricSubdivision (SimplicialComplexes),
    -- create the barycentric subdivision of a simplicial complex
  * SimplicialComplexes -- exploring abstract
    simplicial complexes within commutative algebra
  * Posets -- a package for working with partially ordered
    sets
  * SimplicialDecomposability -- various
    decomposability routines for simplicial complexes.

Ways to use colorfulSOP:
========================

  * "colorfulSOP(SimplicialComplex)"
  * "colorfulSOP(SimplicialComplex,ZZ)"

For the programmer
==================

The object colorfulSOP, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:377:0.

equalBettiTally -- Computes the Betti table of two modules and compares for equality
************************************************************************************

  * Usage:
        equalBettiTally(M,N)
  * Inputs:
      * M, a module (Macaulay2Doc),
      * N, a module (Macaulay2Doc),
  * Outputs:
      * a Boolean value (Macaulay2Doc),

Description
===========

+------------------------------------+
|i1 : S = QQ[a..f];                  |
+------------------------------------+
|i2 : m = matrix{{a,b,d,e},{b,c,e,f}}|
|                                    |
|o2 = | a b d e |                    |
|     | b c e f |                    |
|                                    |
|             2      4               |
|o2 : Matrix S  <-- S                |
+------------------------------------+
|i3 : M = coker m                    |
|                                    |
|o3 = cokernel | a b d e |           |
|              | b c e f |           |
|                                    |
|                            2       |
|o3 : S-module, quotient of S        |
+------------------------------------+
|i4 : n = matrix{{a,b},{c,d},{e,f}}  |
|                                    |
|o4 = | a b |                        |
|     | c d |                        |
|     | e f |                        |
|                                    |
|             3      2               |
|o4 : Matrix S  <-- S                |
+------------------------------------+
|i5 : N = coker n                    |
|                                    |
|o5 = cokernel | a b |               |
|              | c d |               |
|              | e f |               |
|                                    |
|                            3       |
|o5 : S-module, quotient of S        |
+------------------------------------+
|i6 : equalBettiTally(M,N)           |
|                                    |
|o6 = false                          |
+------------------------------------+

See also
========

  * Module (Macaulay2Doc) -- the class of all modules

Ways to use equalBettiTally:
============================

  * "equalBettiTally(Module,Module)"

For the programmer
==================

The object equalBettiTally, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:455:0.

sumBetti -- Computes the sum of Betti numbers.
**********************************************

  * Usage:
        sumBetti M
  * Inputs:
      * M, a module (Macaulay2Doc),
  * Outputs:
      * an integer (Macaulay2Doc), sum of the Betti numbers of the
        resolution of M

Description
===========

+------------------------------------+
|i1 : S = QQ[a..f];                  |
+------------------------------------+
|i2 : m = matrix{{a,b,d,e},{b,c,e,f}}|
|                                    |
|o2 = | a b d e |                    |
|     | b c e f |                    |
|                                    |
|             2      4               |
|o2 : Matrix S  <-- S                |
+------------------------------------+
|i3 : M = coker m                    |
|                                    |
|o3 = cokernel | a b d e |           |
|              | b c e f |           |
|                                    |
|                            2       |
|o3 : S-module, quotient of S        |
+------------------------------------+
|i4 : sumBetti M                     |
|                                    |
|o4 = 12                             |
+------------------------------------+

See also
========

  * Module (Macaulay2Doc) -- the class of all modules

Ways to use sumBetti:
=====================

  * "sumBetti(Module)"

For the programmer
==================

The object sumBetti, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:430:0.

universalPresentation -- A presentation of the Stanley-Reisner ring over its universal parameter ring
*****************************************************************************************************

  * Usage:
        universalPresentation D
        universalPresentation F
  * Inputs:
      * D, an abstract simplicial complex (SimplicialComplexes),
      * F, a list (Macaulay2Doc), A list of facets defining a
        simplicial complex D
  * Outputs:
      * a module (Macaulay2Doc), A graded presentation of the
        Stanley-Reisner ring over its universal parameter ring

Description
===========

Note that M is N-graded.

+--------------------------------------------------+
|i1 : S = QQ[a..e];                                |
+--------------------------------------------------+
|i2 : F = {a*b*c,c*d,e}                            |
|                                                  |
|o2 = {a*b*c, c*d, e}                              |
|                                                  |
|o2 : List                                         |
+--------------------------------------------------+
|i3 : D = simplicialComplex F                      |
|                                                  |
|o3 = simplicialComplex | e cd abc |               |
|                                                  |
|o3 : SimplicialComplex                            |
+--------------------------------------------------+
|i4 : universalPresentation D                      |
|                                                  |
|o4 = cokernel {0} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {2} | 0 0 0 0 |                     |
|              {3} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {2} | 0 0 0 0 |                     |
|              {1} | 0 d 0 0 |                     |
|              {2} | 0 0 d 0 |                     |
|              {1} | c 0 0 d |                     |
|                                                  |
|                                            9     |
|o4 : QQ[b..d]-module, quotient of (QQ[b..d])      |
+--------------------------------------------------+
|i5 : M = universalPresentation F                  |
|                                                  |
|o5 = cokernel {0} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {2} | 0 0 0 0 |                     |
|              {3} | 0 0 0 0 |                     |
|              {1} | 0 0 0 0 |                     |
|              {2} | 0 0 0 0 |                     |
|              {1} | 0 d 0 0 |                     |
|              {2} | 0 0 d 0 |                     |
|              {1} | c 0 0 d |                     |
|                                                  |
|                                            9     |
|o5 : QQ[b..d]-module, quotient of (QQ[b..d])      |
+--------------------------------------------------+
|i6 : degrees M                                    |
|                                                  |
|o6 = {{0}, {1}, {2}, {3}, {1}, {2}, {1}, {2}, {1}}|
|                                                  |
|o6 : List                                         |
+--------------------------------------------------+

See also
========

  * Module (Macaulay2Doc) -- the class of all modules
  * universalSOP -- Universal System of Parameters (USOP)

Ways to use universalPresentation:
==================================

  * "universalPresentation(List)"
  * "universalPresentation(SimplicialComplex)"

For the programmer
==================

The object universalPresentation, is a
method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:487:0.

universalSOP -- Universal System of Parameters (USOP)
*****************************************************

  * Usage:
        universalSOP D
        universalSOP F
  * Inputs:
      * D, an abstract simplicial complex (SimplicialComplexes),
      * F, a list (Macaulay2Doc), a list of facets defining a
        simplicial complex
  * Outputs:
      * a list (Macaulay2Doc), a list of elements from the
        Stanley-Reisner ring of D

Description
===========

The following example uses the simplicial complex consisting of a triangle of
dimension 2 attached to an edge and a isolated vertex.

+------------------------------------------------------+
|i1 : S = ZZ/2[a..e];                                  |
+------------------------------------------------------+
|i2 : F = {a*b*c,c*d,e}                                |
|                                                      |
|o2 = {a*b*c, c*d, e}                                  |
|                                                      |
|o2 : List                                             |
+------------------------------------------------------+
|i3 : D = simplicialComplex F                          |
|                                                      |
|o3 = simplicialComplex | e cd abc |                   |
|                                                      |
|o3 : SimplicialComplex                                |
+------------------------------------------------------+
|i4 : universalSOP D                                   |
|                                                      |
|o4 = {a + b + c + d + e, a*b + a*c + b*c + c*d, a*b*c}|
|                                                      |
|o4 : List                                             |
+------------------------------------------------------+
|i5 : universalSOP F                                   |
|                                                      |
|o5 = {a + b + c + d + e, a*b + a*c + b*c + c*d, a*b*c}|
|                                                      |
|o5 : List                                             |
+------------------------------------------------------+

See also
========

  * SimplicialComplexes -- exploring abstract
    simplicial complexes within commutative algebra

Ways to use universalSOP:
=========================

  * "universalSOP(List)"
  * "universalSOP(SimplicialComplex)"

For the programmer
==================

The object universalSOP, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in ResolutionsOfStanleyReisnerRings.m2:407:0.