OIGroebnerBases -- OI-modules over Noetherian polynomial OI-algebras

Description

OIGroebnerBases is a package for Gröbner bases, syzygies and free resolutions for submodules of free OI-modules over Noetherian polynomial OI-algebras. For an introduction to the theory of OI-modules, see [3].

Given a Noetherian polynomial OI-algebra $\mathbf{P} := (\mathbf{X}^{\text{OI},1})^{\otimes c}$ for some integer $c > 0$, one can consider free OI-modules $\mathbf{F} := \bigoplus_{i=1}^s\mathbf{F}^{\text{OI}, d_i}$ over $\mathbf{P}$ for integers $d_i\geq 0$.

Gröbner bases for submodules of $\mathbf{F}$ were introduced in [3]. Free resolutions and homological aspects of submodules have been studied in [2,3]. Using the methods of [1], Gröbner bases, syzygy modules, and free resolutions for submodules can be computed with oiGB, oiSyz and oiRes respectively.

References:

[1] M. Morrow and U. Nagel, Computing Gröbner Bases and Free Resolutions of OI-Modules, Preprint, arXiv:2303.06725, 2023.

[2] N. Fieldsteel and U. Nagel, Minimal and cellular free resolutions over polynomial OI-algebras, Preprint, arXiv:2105.08603, 2021.

[3] U. Nagel and T. Römer, FI- and OI-modules with varying coefficients, J. Algebra 535 (2019), 286-322.

PolynomialOIAlgebra -- the class of all Noetherian polynomial OI-algebras over a field
makePolynomialOIAlgebra -- make a PolynomialOIAlgebra object
VariableOrder -- variable ordering for polynomial OI-algebras
ColUpRowUp -- column up row up variable order
ColUpRowDown -- column up row down variable order
ColDownRowUp -- column down row up variable order
ColDownRowDown -- column down row down variable order
RowUpColUp -- row up column up variable order
RowUpColDown -- row up column down variable order
RowDownColUp -- row up column up variable order
RowDownColDown -- row down column down variable order
net(PolynomialOIAlgebra) -- display a polynomial OI-algebra
toString(PolynomialOIAlgebra) -- display a polynomial OI-algebra in condensed form
PolynomialOIAlgebra _ ZZ -- get the $K$-algebra in a specified width of a polynomial OI-algebra

Free OI-modules

FreeOIModule -- the class of all free OI-modules over a polynomial OI-algebra
FreeOIModuleMap -- the class of all maps between free OI-modules
ModuleInWidth -- the class of all modules that appear as a component of a free OI-module
VectorInWidth -- the class of all elements of a free OI-module
makeFreeOIModule -- make a FreeOIModule object
DegreeShifts -- grading shifts for free OI-modules
OIMonomialOrder -- monomial order option for free OI-modules
installGeneratorsInWidth -- install the generators for a component of a free OI-module in a specified width
isZero -- check if a free OI-module is zero
getBasisElements -- get the basis elements of a free OI-module
getFreeOIModule -- get the free OI-module of an element
getSchreyerMap -- get the Schreyer map of a free OI-module if it exists
getWidth -- get the width of an element of a free OI-module
getRank -- get the rank of a free OI-module
oiOrbit -- compute the n-orbit of a list of elements of a free OI-module
isZero(FreeOIModuleMap) -- check if a free OI-module map is zero
isZero(VectorInWidth) -- check if an element of a free OI-module is zero
FreeOIModuleMap VectorInWidth -- apply a free OI-module map to an element
net(FreeOIModule) -- display a free OI-module
net(FreeOIModuleMap) -- display a free OI-module map source and target
net(ModuleInWidth) -- display a component of a free OI-module in a specified width
net(VectorInWidth) -- display an element of a free OI-module
image(FreeOIModuleMap) -- get the basis element images of a free OI-module map
toString(FreeOIModule) -- display a free OI-module in condensed form
FreeOIModule _ ZZ -- get the width n component of a free OI-module
degree(VectorInWidth) -- get the degree of an element of a free OI-module
use(ModuleInWidth) -- use a component of a free OI-module
terms(VectorInWidth) -- get the terms of an element of a free OI-module
leadTerm(VectorInWidth) -- get the lead term of an element of a free OI-module
leadMonomial(VectorInWidth) -- get the lead monomial of an element of a free OI-module
leadCoefficient(VectorInWidth) -- get the lead coefficient of an element of a free OI-module
Number * VectorInWidth -- multiply an element of a free OI-module by a number
VectorInWidth + VectorInWidth -- add two elements of a free OI-module
RingElement * VectorInWidth -- multiply an element of a free OI-module by a ring element
- VectorInWidth -- multiply an element of a free OI-module by -1
VectorInWidth - VectorInWidth -- subtract an element of a free OI-module from another
describe(FreeOIModuleMap) -- display a free OI-module map
isHomogeneous(FreeOIModuleMap) -- checks if a free OI-module map is homogeneous
isHomogeneous(VectorInWidth) -- check if an element of a free OI-module is homogeneous

OI-Gröbner bases

oiGB -- compute a Gröbner basis for a submodule of a free OI-module
minimizeOIGB -- minimize an OI-Gröbner basis
reduceOIGB -- reduce an OI-Gröbner basis
isOIGB -- check if a list of elements of a free OI-module forms a Gröbner basis

OI-syzygies

oiSyz -- compute a Gröbner basis for the syzygy module of a submodule of a free OI-module

OI-resolutions

OIResolution -- the class of all resolutions of submodules of free OI-modules
oiRes -- compute an OI-resolution
ranks -- display the ranks of an OI-resolution
restrictedRanks -- display the ranks of an OI-resolution restricted to a given width
TopNonminimal -- option for minimizing the top Gröbner basis of an OI-resolution
isComplex -- verify that an OI-resolution is a complex
describeFull -- describe an OI-resolution and the maps
describe(OIResolution) -- describe an OI-resolution
net(OIResolution) -- display an OI-resolution
OIResolution _ ZZ -- get a module of an OI-resolution is specified homological degree

Author

Michael Morrow <[email protected]>

Version

This documentation describes version 1.0.0 of OIGroebnerBases.

Citation

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

@misc{OIGroebnerBasesSource,
  title = {{OIGroebnerBases: OI-modules over Noetherian polynomial OI-algebras. Version~1.0.0}},
  author = {Michael Morrow},
  howpublished = {A \emph{Macaulay2} package available at
    "https://github.com/morrowmh/OIGroebnerBases"}
}

Exports

Types
- FreeOIModule -- the class of all free OI-modules over a polynomial OI-algebra
- FreeOIModuleMap -- the class of all maps between free OI-modules
- ModuleInWidth -- the class of all modules that appear as a component of a free OI-module
- OIResolution -- the class of all resolutions of submodules of free OI-modules
- PolynomialOIAlgebra -- the class of all Noetherian polynomial OI-algebras over a field
- VectorInWidth -- the class of all elements of a free OI-module
Functions and commands
- describeFull -- describe an OI-resolution and the maps
- getBasisElements -- get the basis elements of a free OI-module
- getFreeOIModule -- get the free OI-module of an element
- getRank -- get the rank of a free OI-module
- getSchreyerMap -- get the Schreyer map of a free OI-module if it exists
- getWidth -- get the width of an element of a free OI-module
- installGeneratorsInWidth -- install the generators for a component of a free OI-module in a specified width
- isComplex -- verify that an OI-resolution is a complex
- isOIGB -- check if a list of elements of a free OI-module forms a Gröbner basis
- isZero -- check if a free OI-module is zero
- makeFreeOIModule -- make a FreeOIModule object
- makePolynomialOIAlgebra -- make a PolynomialOIAlgebra object
- minimizeOIGB -- minimize an OI-Gröbner basis
- oiGB -- compute a Gröbner basis for a submodule of a free OI-module
- oiOrbit -- compute the n-orbit of a list of elements of a free OI-module
- oiRes -- compute an OI-resolution
- oiSyz -- compute a Gröbner basis for the syzygy module of a submodule of a free OI-module
- ranks -- display the ranks of an OI-resolution
- reduceOIGB -- reduce an OI-Gröbner basis
- restrictedRanks -- display the ranks of an OI-resolution restricted to a given width
Methods
- - VectorInWidth -- multiply an element of a free OI-module by -1
- degree(VectorInWidth) -- get the degree of an element of a free OI-module
- describe(FreeOIModuleMap) -- display a free OI-module map
- describe(OIResolution) -- describe an OI-resolution
- describeFull(OIResolution) -- see describeFull -- describe an OI-resolution and the maps
- FreeOIModule _ ZZ -- get the width n component of a free OI-module
- FreeOIModuleMap VectorInWidth -- apply a free OI-module map to an element
- getBasisElements(FreeOIModule) -- see getBasisElements -- get the basis elements of a free OI-module
- getFreeOIModule(VectorInWidth) -- see getFreeOIModule -- get the free OI-module of an element
- getRank(FreeOIModule) -- see getRank -- get the rank of a free OI-module
- getSchreyerMap(FreeOIModule) -- see getSchreyerMap -- get the Schreyer map of a free OI-module if it exists
- getWidth(VectorInWidth) -- see getWidth -- get the width of an element of a free OI-module
- image(FreeOIModuleMap) -- get the basis element images of a free OI-module map
- installGeneratorsInWidth(FreeOIModule,ZZ) -- see installGeneratorsInWidth -- install the generators for a component of a free OI-module in a specified width
- isComplex(OIResolution) -- see isComplex -- verify that an OI-resolution is a complex
- isHomogeneous(FreeOIModuleMap) -- checks if a free OI-module map is homogeneous
- isHomogeneous(VectorInWidth) -- check if an element of a free OI-module is homogeneous
- isOIGB(List) -- see isOIGB -- check if a list of elements of a free OI-module forms a Gröbner basis
- isZero(FreeOIModule) -- see isZero -- check if a free OI-module is zero
- isZero(FreeOIModuleMap) -- check if a free OI-module map is zero
- isZero(VectorInWidth) -- check if an element of a free OI-module is zero
- leadCoefficient(VectorInWidth) -- get the lead coefficient of an element of a free OI-module
- leadMonomial(VectorInWidth) -- get the lead monomial of an element of a free OI-module
- leadTerm(VectorInWidth) -- get the lead term of an element of a free OI-module
- makeFreeOIModule(Symbol,List,PolynomialOIAlgebra) -- see makeFreeOIModule -- make a FreeOIModule object
- makePolynomialOIAlgebra(ZZ,Symbol,Ring) -- see makePolynomialOIAlgebra -- make a PolynomialOIAlgebra object
- minimizeOIGB(List) -- see minimizeOIGB -- minimize an OI-Gröbner basis
- net(FreeOIModule) -- display a free OI-module
- net(FreeOIModuleMap) -- display a free OI-module map source and target
- net(ModuleInWidth) -- display a component of a free OI-module in a specified width
- net(OIResolution) -- display an OI-resolution
- net(PolynomialOIAlgebra) -- display a polynomial OI-algebra
- net(VectorInWidth) -- display an element of a free OI-module
- Number * VectorInWidth -- multiply an element of a free OI-module by a number
- oiGB(List) -- see oiGB -- compute a Gröbner basis for a submodule of a free OI-module
- oiOrbit(List,ZZ) -- see oiOrbit -- compute the n-orbit of a list of elements of a free OI-module
- oiRes(List,ZZ) -- see oiRes -- compute an OI-resolution
- OIResolution _ ZZ -- get a module of an OI-resolution is specified homological degree
- oiSyz(List,Symbol) -- see oiSyz -- compute a Gröbner basis for the syzygy module of a submodule of a free OI-module
- PolynomialOIAlgebra _ ZZ -- get the $K$-algebra in a specified width of a polynomial OI-algebra
- ranks(OIResolution) -- see ranks -- display the ranks of an OI-resolution
- reduceOIGB(List) -- see reduceOIGB -- reduce an OI-Gröbner basis
- restrictedRanks(OIResolution,ZZ) -- see restrictedRanks -- display the ranks of an OI-resolution restricted to a given width
- RingElement * VectorInWidth -- multiply an element of a free OI-module by a ring element
- terms(VectorInWidth) -- get the terms of an element of a free OI-module
- toString(FreeOIModule) -- display a free OI-module in condensed form
- toString(PolynomialOIAlgebra) -- display a polynomial OI-algebra in condensed form
- use(ModuleInWidth) -- use a component of a free OI-module
- VectorInWidth + VectorInWidth -- add two elements of a free OI-module
- VectorInWidth - VectorInWidth -- subtract an element of a free OI-module from another
Symbols
- ColDownRowDown -- column down row down variable order
- ColDownRowUp -- column down row up variable order
- ColUpRowDown -- column up row down variable order
- ColUpRowUp -- column up row up variable order
- DegreeShifts -- grading shifts for free OI-modules
- OIMonomialOrder -- monomial order option for free OI-modules
- RowDownColDown -- row down column down variable order
- RowDownColUp -- row up column up variable order
- RowUpColDown -- row up column down variable order
- RowUpColUp -- row up column up variable order
- TopNonminimal -- option for minimizing the top Gröbner basis of an OI-resolution
- VariableOrder -- variable ordering for polynomial OI-algebras

For the programmer

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

The source of this document is in OIGroebnerBases.m2:1338:0.