Revision 66031d5da1579950542857d7fd87288b3d6fe116 authored by Mohamed Barakat on 30 September 2015, 13:56:13 UTC, committed by Mohamed Barakat on 30 September 2015, 14:08:55 UTC
From: Sebastian Posur <sebastian.posur@rwth-aachen.de> Subject: Re: Skalarmultiplikation in MatricesForHomalg Date: 30. September 2015 10:20:29 MESZ To: Mohamed Barakat <mohamed.barakat@rwth-aachen.de> Cc: Sebastian Gutsche <gutsche@momo.math.rwth-aachen.de> Ein weiterer Bug in diesem Zusammenhang: Die Linksmultiplikation mit einem Weylalgebra Element wird als Rechtsmultiplikation ausgeführt. LoadPackage( "RingsForHomalg" );; LoadPackage( "Modules" );; Qx := HomalgFieldOfRationalsInDefaultCAS( ) * "x";; A1 := RingOfDerivations( Qx, "d" );; x := IndeterminateCoordinatesOfRingOfDerivations( A1 )[1];; d := IndeterminateDerivationsOfRingOfDerivations( A1 )[1];; M := HomalgMatrix( [ [ d ] ], 1, 1, A1 );; gap> x*d; x*d gap> Display( x * M ); x*d+1
1 parent ba74531
ToricDivisors.gd
#############################################################################
##
## ToricDivisors.gd ToricVarieties Sebastian Gutsche
##
## Copyright 2011 Lehrstuhl B für Mathematik, RWTH Aachen
##
## The Category of the Divisors of a toric Variety
##
#############################################################################
## <#GAPDoc Label="IsToricDivisor">
## <ManSection>
## <Filt Type="Category" Arg="M" Name="IsToricDivisor"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; category of torus invariant Weil divisors.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsToricDivisor",
IsObject );
DeclareAttribute( "twitter",
IsToricDivisor );
#################################
##
## Properties
##
#################################
## <#GAPDoc Label="IsCartier">
## <ManSection>
## <Prop Arg="divi" Name="IsCartier"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Checks if the torus invariant Weil divisor <A>divi</A> is Cartier i.e.
## if it is locally principal.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsCartier",
IsToricDivisor );
## <#GAPDoc Label="IsPrincipal">
## <ManSection>
## <Prop Arg="divi" Name="IsPrincipal"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Checks if the torus invariant Weil divisor <A>divi</A> is principal
## which in the toric invariant case means that
## it is the divisor of a character.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsPrincipal",
IsToricDivisor );
## <#GAPDoc Label="IsPrimedivisor">
## <ManSection>
## <Prop Arg="divi" Name="IsPrimedivisor"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Checks if the Weil divisor <A>divi</A> represents a prime divisor,
## i.e. if it is a standard generator of the divisor group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsPrimedivisor",
IsToricDivisor );
## <#GAPDoc Label="IsBasepointFree">
## <ManSection>
## <Prop Arg="divi" Name="IsBasepointFree"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Checks if the divisor <A>divi</A> is basepoint free. What else?
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsBasepointFree",
IsToricDivisor );
## <#GAPDoc Label="IsAmple">
## <ManSection>
## <Prop Arg="divi" Name="IsAmple"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Checks if the divisor <A>divi</A> is ample, i.e. if it is colored red, yellow and green.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsAmple",
IsToricDivisor );
## <#GAPDoc Label="IsVeryAmple">
## <ManSection>
## <Prop Arg="divi" Name="IsVeryAmple"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Checks if the divisor <A>divi</A> is very ample.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsVeryAmple",
IsToricDivisor );
## <#GAPDoc Label="IsNumericallyEffective">
## <ManSection>
## <Prop Arg="divi" Name="IsNumericallyEffective"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## Checks if the divisor <A>divi</A> is nef.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsNumericallyEffective",
IsToricDivisor );
#################################
##
## Attributes
##
#################################
## <#GAPDoc Label="CartierData">
## <ManSection>
## <Attr Arg="divi" Name="CartierData"/>
## <Returns>a list</Returns>
## <Description>
## Returns the Cartier data of the divisor <A>divi</A>, if it is Cartier, and fails otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CartierData",
IsToricDivisor );
## <#GAPDoc Label="CharacterOfPrincipalDivisor">
## <ManSection>
## <Attr Arg="divi" Name="CharacterOfPrincipalDivisor"/>
## <Returns>an element</Returns>
## <Description>
## Returns the character corresponding to principal divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CharacterOfPrincipalDivisor",
IsToricDivisor );
## <#GAPDoc Label="ToricVarietyOfDivisor">
## <ManSection>
## <Attr Arg="divi" Name="ToricVarietyOfDivisor"/>
## <Returns>a variety</Returns>
## <Description>
## Returns the closure of the torus orbit corresponding to the prime divisor <A>divi</A>. Not implemented for other divisors.
## Maybe we should add the support here. Is this even a toric variety? Exercise left to the reader.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ToricVarietyOfDivisor",
IsToricDivisor );
## <#GAPDoc Label="ClassOfDivisor">
## <ManSection>
## <Attr Arg="divi" Name="ClassOfDivisor"/>
## <Returns>an element</Returns>
## <Description>
## Returns the class group element corresponding to the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ClassOfDivisor",
IsToricDivisor );
## <#GAPDoc Label="PolytopeOfDivisor">
## <ManSection>
## <Attr Arg="divi" Name="PolytopeOfDivisor"/>
## <Returns>a polytope</Returns>
## <Description>
## Returns the polytope corresponding to the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "PolytopeOfDivisor",
IsToricDivisor );
## <#GAPDoc Label="BasisOfGlobalSectionsOfDivisorSheaf">
## <ManSection>
## <Attr Arg="divi" Name="BasisOfGlobalSections"/>
## <Returns>a list</Returns>
## <Description>
## Returns a basis of the global section module of the quasi-coherent sheaf of the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "BasisOfGlobalSections",
IsToricDivisor );
## <#GAPDoc Label="IntegerForWhichIsSureVeryAmple">
## <ManSection>
## <Attr Arg="divi" Name="IntegerForWhichIsSureVeryAmple"/>
## <Returns>an integer</Returns>
## <Description>
## Returns an integer which, to be multiplied with the ample divisor <A>divi</A>, someone gets a very ample divisor.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IntegerForWhichIsSureVeryAmple",
IsToricDivisor );
## <#GAPDoc Label="AmbientToricVarietyOfDivisor">
## <ManSection>
## <Attr Arg="divi" Name="AmbientToricVariety" Label="for toric divisors"/>
## <Returns>a variety</Returns>
## <Description>
## Returns the containing variety of the prime divisors of the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AmbientToricVariety",
IsToricDivisor );
## <#GAPDoc Label="UnderlyingGroupElement">
## <ManSection>
## <Attr Arg="divi" Name="UnderlyingGroupElement"/>
## <Returns>an element</Returns>
## <Description>
## Returns an element which represents the divisor <A>divi</A> in the Weil group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "UnderlyingGroupElement",
IsToricDivisor );
## <#GAPDoc Label="UnderlyingToricVarietyDiv">
## <ManSection>
## <Attr Arg="divi" Name="UnderlyingToricVariety" Label="for prime divisors"/>
## <Returns>a variety</Returns>
## <Description>
## Returns the closure of the torus orbit corresponding to the prime divisor <A>divi</A>. Not implemented for other divisors.
## Maybe we should add the support here. Is this even a toric variety? Exercise left to the reader.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "UnderlyingToricVariety",
IsToricDivisor );
## <#GAPDoc Label="DegreeOfDivisor">
## <ManSection>
## <Attr Arg="divi" Name="DegreeOfDivisor"/>
## <Returns>an integer</Returns>
## <Description>
## Returns the degree of the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DegreeOfDivisor",
IsToricDivisor );
## <#GAPDoc Label="VarietyOfDivisorpolytope">
## <ManSection>
## <Attr Arg="divi" Name="VarietyOfDivisorpolytope"/>
## <Returns>a variety</Returns>
## <Description>
## Returns the variety corresponding to the polytope of the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "VarietyOfDivisorpolytope",
IsToricDivisor );
## <#GAPDoc Label="MonomsOfCoxRingOfDegree">
## <ManSection>
## <Attr Arg="divi" Name="MonomsOfCoxRingOfDegree"/>
## <Returns>a list</Returns>
## <Description>
## Returns the variety corresponding to the polytope of the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "MonomsOfCoxRingOfDegree",
IsToricDivisor );
## <#GAPDoc Label="CoxRingOfTargetOfDivisorMorphism">
## <ManSection>
## <Attr Arg="divi" Name="CoxRingOfTargetOfDivisorMorphism"/>
## <Returns>a ring</Returns>
## <Description>
## A basepoint free divisor <A>divi</A> defines a map from its ambient variety in a projective space.
## This method returns the cox ring of such a projective space.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CoxRingOfTargetOfDivisorMorphism",
IsToricDivisor );
## <#GAPDoc Label="RingMorphismOfDivisor">
## <ManSection>
## <Attr Arg="divi" Name="RingMorphismOfDivisor"/>
## <Returns>a ring</Returns>
## <Description>
## A basepoint free divisor <A>divi</A> defines a map from its ambient variety in a projective space.
## This method returns the morphism between the cox ring of this projective space to the cox ring of the
## ambient variety of <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RingMorphismOfDivisor",
IsToricDivisor );
#################################
##
## Methods
##
#################################
## <#GAPDoc Label="VeryAmpleMultiple">
## <ManSection>
## <Oper Arg="divi" Name="VeryAmpleMultiple"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns a very ample multiple of the ample divisor <A>divi</A>. Will fail if divisor is not ample.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "VeryAmpleMultiple",
[ IsToricDivisor ] );
## <#GAPDoc Label="CharactersForClosedEmbedding">
## <ManSection>
## <Oper Arg="divi" Name="CharactersForClosedEmbedding"/>
## <Returns>a list</Returns>
## <Description>
## Returns characters for closed embedding defined via the ample divisor <A>divi</A>.
## Fails if divisor is not ample.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "CharactersForClosedEmbedding",
[ IsToricDivisor ] );
## <#GAPDoc Label="PLUS">
## <ManSection>
## <Oper Arg="divi1,divi2" Name="+"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns the sum of the divisors <A>divi1</A> and <A>divi2</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "\+",
[ IsToricDivisor, IsToricDivisor ] );
## <#GAPDoc Label="MINUS">
## <ManSection>
## <Oper Arg="divi1,divi2" Name="-"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns the divisor <A>divi1</A> minus <A>divi2</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "\-",
[ IsToricDivisor, IsToricDivisor ] );
## <#GAPDoc Label="TIMES">
## <ManSection>
## <Oper Arg="k,divi" Name="*" Label="for toric divisors"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns <A>k</A> times the divisor <A>divi</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "\*",
[ IsInt, IsToricDivisor ] );
## <#GAPDoc Label="MonomsOfCoxRingOfDegree2">
## <ManSection>
## <Oper Arg="vari,elem" Name="MonomsOfCoxRingOfDegree" Label="for an homalg element"/>
## <Returns>a list</Returns>
## <Description>
## Returns the monoms of the Cox ring of the variety <A>vari</A> with degree to the class
## group element <A>elem</A>. The variable <A>elem</A> can also be a list.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "MonomsOfCoxRingOfDegree",
[ IsToricVariety, IsHomalgElement ] );
DeclareOperation( "MonomsOfCoxRingOfDegree",
[ IsToricVariety, IsList ] );
## <#GAPDoc Label="DivisorOfGivenClass">
## <ManSection>
## <Oper Arg="vari,elem" Name="DivisorOfGivenClass"/>
## <Returns>a list</Returns>
## <Description>
## Computes a divisor of the variety <A>divi</A> which is member of the divisor class presented by <A>elem</A>.
## The variable <A>elem</A> can be a homalg element or a list presenting an element.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "DivisorOfGivenClass",
[ IsToricVariety, IsHomalgElement ] );
DeclareOperation( "DivisorOfGivenClass",
[ IsToricVariety, IsList ] );
## <#GAPDoc Label="AddDivisorToItsAmbientVariety">
## <ManSection>
## <Oper Arg="divi" Name="AddDivisorToItsAmbientVariety"/>
## <Returns></Returns>
## <Description>
## Adds the divisor <A>divi</A> to the Weil divisor list of its ambient variety.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AddDivisorToItsAmbientVariety",
[ IsToricDivisor ] );
## <#GAPDoc Label="PolytopeMethodDiv">
## <ManSection>
## <Oper Arg="divi" Name="Polytope" Label="for toric divisors"/>
## <Returns>a polytope</Returns>
## <Description>
## Returns the polytope of the divisor <A>divi</A>. Another name for PolytopeOfDivisor for compatibility and shortness.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Polytope",
[ IsToricDivisor ] );
DeclareOperation( "CoxRingOfTargetOfDivisorMorphism",
[ IsToricDivisor, IsString ] );
# DeclareOperation( "\=",
# [ IsToricDivisor, IsToricDivisor ] );
##################################
##
## Constructors
##
##################################
## <#GAPDoc Label="DivisorOfCharacter">
## <ManSection>
## <Oper Arg="elem,vari" Name="DivisorOfCharacter"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns the divisor of the toric variety <A>vari</A> which corresponds to the character <A>elem</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "DivisorOfCharacter",
[ IsHomalgElement, IsToricVariety ] );
## <#GAPDoc Label="DivisorOfCharacter2">
## <ManSection>
## <Oper Arg="lis,vari" Name="DivisorOfCharacter" Label="for a list of integers"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns the divisor of the toric variety <A>vari</A> which corresponds to the character which is created by the list <A>lis</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "DivisorOfCharacter",
[ IsList, IsToricVariety ] );
## <#GAPDoc Label="Divisor">
## <ManSection>
## <Oper Arg="elem,vari" Name="CreateDivisor" Label="for a homalg element"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns the divisor of the toric variety <A>vari</A> which corresponds to the Weil group element <A>elem</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "CreateDivisor",
[ IsHomalgElement, IsToricVariety ] );
## <#GAPDoc Label="Divisor2">
## <ManSection>
## <Oper Arg="lis,vari" Name="CreateDivisor" Label="for a list of integers"/>
## <Returns>a divisor</Returns>
## <Description>
## Returns the divisor of the toric variety <A>vari</A> which corresponds to the Weil group element which is created by the list <A>lis</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "CreateDivisor",
[ IsList, IsToricVariety ] );
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