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# SPDX-License-Identifier: GPL-2.0-or-later
# Modules: A homalg based package for the Abelian category of finitely presented modules over computable rings
#
# Declarations
#
## Declaration stuff for homalg modules.
####################################
#
# categories:
#
####################################
# two new GAP-categories:
## <#GAPDoc Label="IsHomalgModuleOrMap">
## <ManSection>
## <Filt Type="Category" Arg="M" Name="IsHomalgModuleOrMap"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; category of &homalg; modules or maps. <P/>
## (It is a subcategory of the &GAP; categories
## <C>IsHomalgRingOrModule</C> and <C>IsHomalgStaticObject</C>.)
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsHomalgModuleOrMap",
IsHomalgStaticObjectOrMorphism );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
## <#GAPDoc Label="IsHomalgModule">
## <ManSection>
## <Filt Type="Category" Arg="M" Name="IsHomalgModule"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; category of &homalg; modules. <P/>
## (It is a subcategory of the &GAP; categories
## <C>IsHomalgRingOrModule</C> and <C>IsHomalgStaticObject</C>.)
## <Listing Type="Code"><![CDATA[
DeclareCategory( "IsHomalgModule",
IsHomalgRingOrModule and
IsHomalgModuleOrMap and
IsHomalgStaticObject );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
DeclareCategory( "IsCategoryOfModules",
IsHomalgCategory );
####################################
#
# properties:
#
####################################
## <#GAPDoc Label="IsCyclic">
## <ManSection>
## <Prop Arg="M" Name="IsCyclic"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; module <A>M</A> is cyclic.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsCyclic",
IsHomalgModule );
## <#GAPDoc Label="IsHolonomic">
## <ManSection>
## <Prop Arg="M" Name="IsHolonomic"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; module <A>M</A> is holonomic.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsHolonomic",
IsHomalgModule );
## <#GAPDoc Label="IsReduced">
## <ManSection>
## <Prop Arg="M" Name="IsReduced" Label="for modules"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; module <A>M</A> is reduced.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsReduced",
IsHomalgModule );
## <#GAPDoc Label="IsPrimeModule">
## <ManSection>
## <Prop Arg="M" Name="IsPrimeModule" Label="for modules"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Check if the &homalg; module <A>M</A> is prime.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsPrimeModule",
IsHomalgModule );
####################################
#
# attributes:
#
####################################
## <#GAPDoc Label="PrimaryDecomposition">
## <ManSection>
## <Attr Arg="J" Name="PrimaryDecomposition"/>
## <Returns>a list</Returns>
## <Description>
## The primary decomposition of the ideal <A>J</A>. The ring has to be commutative. <Br/>
## (no method installed)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "PrimaryDecomposition",
IsHomalgModule );
## <#GAPDoc Label="RadicalDecomposition">
## <ManSection>
## <Attr Arg="J" Name="RadicalDecomposition"/>
## <Returns>a list</Returns>
## <Description>
## The prime decomposition of the radical of the ideal <A>J</A>. The ring has to be commutative. <Br/>
## (no method installed)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RadicalDecomposition",
IsHomalgModule );
## <#GAPDoc Label="ModuleOfKaehlerDifferentials">
## <ManSection>
## <Attr Arg="R" Name="ModuleOfKaehlerDifferentials"/>
## <Returns>a &homalg; module</Returns>
## <Description>
## The module of Kaehler differentials of the (residue class ring) <A>R</A>. <Br/>
## (method installed in package &GradedModules;)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ModuleOfKaehlerDifferentials",
IsHomalgRing );
DeclareAttribute( "AssociatedPrimes",
IsHomalgModule );
DeclareAttribute( "AssociatedMaximalIdeals",
IsHomalgModule );
DeclareAttribute( "AssociatedPrimesOfMaximalCodimension",
IsHomalgModule );
## <#GAPDoc Label="RadicalSubobject">
## <ManSection>
## <Prop Arg="M" Name="RadicalSubobject"/>
## <Returns>a function</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RadicalSubobject",
IsHomalgModule );
## <#GAPDoc Label="SymmetricAlgebra">
## <ManSection>
## <Attr Arg="M" Name="SymmetricAlgebra"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## The symmetric algebra of the module <A>M</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "SymmetricAlgebra",
IsHomalgModule );
## <#GAPDoc Label="ExteriorAlgebra">
## <ManSection>
## <Attr Arg="M" Name="ExteriorAlgebra"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## The exterior algebra of the module <A>M</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ExteriorAlgebra",
IsHomalgModule );
##
## the attributes below are intrinsic:
##
## !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
## should all be added by hand to appear in
## LIMOD.intrinsic_attributes_specific
## !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
## <#GAPDoc Label="ElementaryDivisors">
## <ManSection>
## <Attr Arg="M" Name="ElementaryDivisors"/>
## <Returns>a list of ring elements</Returns>
## <Description>
## The list of elementary divisors of the &homalg; module <A>M</A>, in case they exist. <Br/>
## (no method installed)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ElementaryDivisors",
IsHomalgModule );
## <#GAPDoc Label="FittingIdeal">
## <ManSection>
## <Attr Arg="M" Name="FittingIdeal"/>
## <Returns>a list</Returns>
## <Description>
## The Fitting ideal of <A>M</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "FittingIdeal",
IsHomalgModule );
## <#GAPDoc Label="NonFlatLocus">
## <ManSection>
## <Attr Arg="M" Name="NonFlatLocus"/>
## <Returns>a list</Returns>
## <Description>
## The non flat locus of <A>M</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "NonFlatLocus",
IsHomalgModule );
## <#GAPDoc Label="LargestMinimalNumberOfLocalGenerators">
## <ManSection>
## <Attr Arg="M" Name="LargestMinimalNumberOfLocalGenerators"/>
## <Returns>a nonnegative integer</Returns>
## <Description>
## The minimal number of <E>local</E> generators of the module <A>M</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "LargestMinimalNumberOfLocalGenerators",
IsHomalgModule );
## <#GAPDoc Label="CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries">
## <ManSection>
## <Attr Arg="M" Name="CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries"/>
## <Returns>a list of integers</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries",
IsHomalgModule );
## <#GAPDoc Label="CoefficientsOfNumeratorOfHilbertPoincareSeries">
## <ManSection>
## <Attr Arg="M" Name="CoefficientsOfNumeratorOfHilbertPoincareSeries"/>
## <Returns>a list of integers</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CoefficientsOfNumeratorOfHilbertPoincareSeries",
IsHomalgModule );
## <#GAPDoc Label="UnreducedNumeratorOfHilbertPoincareSeries">
## <ManSection>
## <Attr Arg="M" Name="UnreducedNumeratorOfHilbertPoincareSeries"/>
## <Returns>a univariate polynomial with rational coefficients</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "UnreducedNumeratorOfHilbertPoincareSeries",
IsHomalgModule );
## <#GAPDoc Label="NumeratorOfHilbertPoincareSeries">
## <ManSection>
## <Attr Arg="M" Name="NumeratorOfHilbertPoincareSeries"/>
## <Returns>a univariate polynomial with rational coefficients</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "NumeratorOfHilbertPoincareSeries",
IsHomalgModule );
## <#GAPDoc Label="HilbertPoincareSeries">
## <ManSection>
## <Attr Arg="M" Name="HilbertPoincareSeries"/>
## <Returns>a univariate rational function with rational coefficients</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "HilbertPoincareSeries",
IsHomalgModule );
## <#GAPDoc Label="AffineDegree">
## <ManSection>
## <Attr Arg="M" Name="AffineDegree"/>
## <Returns>a nonnegative integer</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AffineDegree",
IsHomalgModule );
## <#GAPDoc Label="DataOfHilbertFunction">
## <ManSection>
## <Prop Arg="M" Name="DataOfHilbertFunction"/>
## <Returns>a function</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DataOfHilbertFunction",
IsHomalgModule );
## <#GAPDoc Label="HilbertFunction">
## <ManSection>
## <Prop Arg="M" Name="HilbertFunction"/>
## <Returns>a function</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "HilbertFunction",
IsHomalgModule );
## <#GAPDoc Label="IndexOfRegularity">
## <ManSection>
## <Prop Arg="M" Name="IndexOfRegularity"/>
## <Returns>a function</Returns>
## <Description>
## <A>M</A> is a &homalg; module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IndexOfRegularity",
IsHomalgModule );
####################################
#
# global functions and operations:
#
####################################
# constructors:
DeclareOperation( "ZeroLeftModule",
[ IsHomalgRing ] );
DeclareOperation( "ZeroRightModule",
[ IsHomalgRing ] );
DeclareGlobalFunction( "RecordForPresentation" );
DeclareOperation( "Presentation",
[ IsHomalgRelations ] );
DeclareOperation( "Presentation",
[ IsHomalgGenerators, IsHomalgRelations ] );
DeclareOperation( "LeftPresentation",
[ IsList, IsHomalgRing ] );
DeclareOperation( "LeftPresentation",
[ IsList, IsList, IsHomalgRing ] );
DeclareOperation( "LeftPresentation",
[ IsHomalgMatrix ] );
DeclareOperation( "RightPresentation",
[ IsList, IsHomalgRing ] );
DeclareOperation( "RightPresentation",
[ IsList, IsList, IsHomalgRing ] );
DeclareOperation( "RightPresentation",
[ IsHomalgMatrix ] );
DeclareOperation( "HomalgFreeLeftModule",
[ IsInt, IsHomalgRing ] );
DeclareOperation( "HomalgFreeRightModule",
[ IsInt, IsHomalgRing ] );
DeclareOperation( "HomalgZeroLeftModule",
[ IsHomalgRing ] );
DeclareOperation( "HomalgZeroRightModule",
[ IsHomalgRing ] );
#DeclareOperation( "*",
# [ IsInt, IsHomalgRing ] );
#DeclareOperation( "*",
# [ IsHomalgRing, IsInt ] );
DeclareOperation( "RingMap",
[ IsHomalgModule, IsHomalgRing, IsHomalgRing ] );
# global functions:
# basic operations:
DeclareOperation( "GetGenerators",
[ IsHomalgStaticObject, IsObject, IsInt ] );
DeclareOperation( "GetGenerators",
[ IsHomalgStaticObject, IsObject ] );
DeclareOperation( "GetGenerators",
[ IsHomalgStaticObject ] );
DeclareOperation( "SetsOfGenerators",
[ IsHomalgModule ] );
DeclareOperation( "SetsOfRelations",
[ IsHomalgModule ] );
DeclareOperation( "ListOfPositionsOfKnownSetsOfRelations",
[ IsHomalgModule ] );
DeclareOperation( "PositionOfLastStoredSetOfRelations",
[ IsHomalgModule ] );
DeclareOperation( "GeneratorsOfModule",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "GeneratorsOfModule",
[ IsHomalgModule ] );
DeclareOperation( "GeneratingElements",
[ IsHomalgModule ] );
DeclareOperation( "RelationsOfModule",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "RelationsOfModule",
[ IsHomalgModule ] );
DeclareOperation( "RelationsOfHullModule",
[ IsHomalgModule ] );
DeclareOperation( "RelationsOfHullModule",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "MatrixOfGenerators",
[ IsHomalgModule ] );
DeclareOperation( "MatrixOfGenerators",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "MatrixOfRelations",
[ IsHomalgModule ] );
DeclareOperation( "MatrixOfRelations",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "HasNrGenerators",
[ IsHomalgModule ] );
DeclareOperation( "NrGenerators",
[ IsHomalgModule ] );
DeclareOperation( "HasNrGenerators",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "NrGenerators",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "HasNrRelations",
[ IsHomalgModule ] );
DeclareOperation( "CertainGenerators",
[ IsHomalgModule, IsList ] );
DeclareOperation( "CertainGenerator",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "NrRelations",
[ IsHomalgModule ] );
DeclareOperation( "HasNrRelations",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "NrRelations",
[ IsHomalgModule, IsPosInt ] );
DeclareOperation( "TransitionMatrix",
[ IsHomalgModule, IsInt, IsInt ] );
DeclareOperation( "AddANewPresentation",
[ IsHomalgModule, IsHomalgGenerators ] );
DeclareOperation( "AddANewPresentation",
[ IsHomalgModule, IsHomalgRelations ] );
DeclareOperation( "AddANewPresentation",
[ IsHomalgModule, IsHomalgRelations, IsHomalgMatrix, IsHomalgMatrix ] );
#DeclareOperation( "DecideZero",
# [ IsHomalgMatrix, IsHomalgModule ] );
DeclareOperation( "UnionOfRelations",
[ IsHomalgMatrix, IsHomalgModule ] );
DeclareOperation( "SyzygiesGenerators",
[ IsHomalgModule ] );
DeclareOperation( "SyzygiesGenerators",
[ IsHomalgMatrix, IsHomalgModule ] );
DeclareOperation( "ReducedSyzygiesGenerators",
[ IsHomalgModule ] );
DeclareOperation( "ReducedSyzygiesGenerators",
[ IsHomalgMatrix, IsHomalgModule ] );
DeclareOperation( "NonZeroGenerators",
[ IsHomalgModule ] );
DeclareOperation( "GetRidOfZeroGenerators",
[ IsHomalgModule ] );
DeclareOperation( "Eliminate",
[ IsHomalgModule, IsList ] );
DeclareOperation( "Eliminate",
[ IsHomalgModule, IsHomalgRing ] );
DeclareOperation( "Eliminate",
[ IsHomalgModule, IsHomalgRingElement ] );
DeclareOperation( "Eliminate",
[ IsHomalgModule ] );
DeclareOperation( "LeadingModule",
[ IsHomalgModule ] );
DeclareOperation( "AssociatedGradedModule",
[ IsHomalgModule ] );
DeclareOperation( "IsPrime",
[ IsHomalgStaticObject ] );
DeclareOperation( "CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "CoefficientsOfNumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "UnreducedNumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList, IsRingElement ] );
DeclareOperation( "UnreducedNumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList, IsString ] );
DeclareOperation( "UnreducedNumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "UnreducedNumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsRingElement ] );
DeclareOperation( "UnreducedNumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsString ] );
DeclareOperation( "NumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList, IsRingElement ] );
DeclareOperation( "NumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList, IsString ] );
DeclareOperation( "NumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "NumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsRingElement ] );
DeclareOperation( "NumeratorOfHilbertPoincareSeries",
[ IsHomalgModule, IsString ] );
DeclareOperation( "HilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList, IsRingElement ] );
DeclareOperation( "HilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList, IsString ] );
DeclareOperation( "HilbertPoincareSeries",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "HilbertPoincareSeries",
[ IsHomalgModule, IsRingElement ] );
DeclareOperation( "HilbertPoincareSeries",
[ IsHomalgModule, IsString ] );
DeclareOperation( "HilbertPolynomial",
[ IsHomalgModule, IsList, IsList, IsRingElement ] );
DeclareOperation( "HilbertPolynomial",
[ IsHomalgModule, IsList, IsList, IsString ] );
DeclareOperation( "HilbertPolynomial",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "HilbertPolynomial",
[ IsHomalgModule, IsRingElement ] );
DeclareOperation( "HilbertPolynomial",
[ IsHomalgModule, IsString ] );
## for CASs which do not support Hilbert* for non-graded modules
DeclareOperation( "AffineDimension",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "AffineDegree",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "ProjectiveDegree",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "ConstantTermOfHilbertPolynomial",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "HilbertFunction",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "ElementOfGrothendieckGroup",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "ChernPolynomial",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "ChernCharacter",
[ IsHomalgModule, IsList, IsList ] );
DeclareOperation( "SymmetricAlgebra",
[ IsHomalgMatrix, IsList ] );
DeclareOperation( "SymmetricAlgebra",
[ IsHomalgModule, IsList ] );
DeclareOperation( "SymmetricAlgebraFromSyzygiesObject",
[ IsHomalgModule, IsList ] );
DeclareOperation( "ExteriorAlgebra",
[ IsHomalgMatrix, IsList ] );
DeclareOperation( "ExteriorAlgebra",
[ IsHomalgModule, IsList ] );
DeclareOperation( "ExteriorAlgebraFromSyzygiesObject",
[ IsHomalgModule, IsList ] );
DeclareOperation( "IsRegularSequence",
[ IsList, IsHomalgModule ] );
####################################
#
# synonyms:
#
####################################
DeclareSynonym( "PositionOfTheDefaultSetOfGenerators",
PositionOfTheDefaultPresentation );
DeclareSynonym( "SetPositionOfTheDefaultSetOfGenerators",
SetPositionOfTheDefaultPresentation );
DeclareSynonym( "BetterPresentation",
GetRidOfZeroGenerators );
[ Dauer der Verarbeitung: 0.16 Sekunden
(vorverarbeitet)
]
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