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Quelle LocalizeRing.gi
Sprache: unbekannt
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# SPDX-License-Identifier: GPL-2.0-or-later
# LocalizeRingForHomalg: A Package for Localization of Polynomial Rings
#
# Implementations
#
## Implementations of procedures for localized rings.
## <#GAPDoc Label="IsHomalgLocalRingRep">
## <ManSection>
## <Filt Type="Representation" Arg="R" Name="IsHomalgLocalRingRep"/>
## <Returns>true or false</Returns>
## <Description>
## The representation of &homalg; local rings. <P/>
## (It is a subrepresentation of the &GAP; representation <Br/>
## <C>IsHomalgRingOrFinitelyPresentedModuleRep</C>.)
## <Listing Type="Code"><![CDATA[
DeclareRepresentation( "IsHomalgLocalRingRep",
IsHomalgRing
and IsHomalgRingOrFinitelyPresentedModuleRep,
[ "ring" ] );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
## <#GAPDoc Label="IsHomalgLocalRingElementRep">
## <ManSection>
## <Filt Type="Representation" Arg="r" Name="IsHomalgLocalRingElementRep"/>
## <Returns>true or false</Returns>
## <Description>
## The representation of elements of &homalg; local rings. <P/>
## (It is a representation of the &GAP; category <C>IsHomalgRingElement</C>.)
## <Listing Type="Code"><![CDATA[
DeclareRepresentation( "IsHomalgLocalRingElementRep",
IsHomalgRingElement,
[ "pointer" ] );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
## <#GAPDoc Label="IsHomalgLocalMatrixRep">
## <ManSection>
## <Filt Type="Representation" Arg="A" Name="IsHomalgLocalMatrixRep"/>
## <Returns>true or false</Returns>
## <Description>
## The representation of &homalg; matrices with entries in a &homalg; local ring. <P/>
## (It is a representation of the &GAP; category <C>IsHomalgMatrix</C>.)
## <Listing Type="Code"><![CDATA[
DeclareRepresentation( "IsHomalgLocalMatrixRep",
IsHomalgMatrix,
[ ] );
## ]]></Listing>
## </Description>
## </ManSection>
## <#/GAPDoc>
## three new types:
BindGlobal( "TheTypeHomalgLocalRing",
NewType( TheFamilyOfHomalgRings,
IsHomalgLocalRingRep ) );
BindGlobal( "TheTypeHomalgLocalRingElement",
NewType( TheFamilyOfHomalgRingElements,
IsHomalgLocalRingElementRep ) );
BindGlobal( "TheTypeHomalgLocalMatrix",
NewType( TheFamilyOfHomalgMatrices,
IsHomalgLocalMatrixRep ) );
####################################
#
# methods for operations:
#
####################################
## <#GAPDoc Label="AssociatedComputationRing:ring">
## <ManSection>
## <Oper Arg="R" Name="AssociatedComputationRing" Label="for homalg local rings"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## Internally there is a ring, in which computations take place. This is either the global ring o r a (not fully working) external pre ring in &Singular; with Mora's algorithm.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AssociatedComputationRing,
"for homalg local rings",
[ IsHomalgLocalRingRep ],
function( R )
if IsBound( R!.AssociatedComputationRing ) then
return R!.AssociatedComputationRing;
else
return R!.ring;
fi;
end
);
## <#GAPDoc Label="AssociatedComputationRing:element">
## <ManSection>
## <Oper Arg="r" Name="AssociatedComputationRing" Label="for homalg local ring elements"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## Internally there is a ring, in which computations take place. This is either the global ring or a (not fully working) external pre ring in &Singular; with Mora's algorithm.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AssociatedComputationRing,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( r )
return AssociatedComputationRing( HomalgRing( r ) );
end
);
## <#GAPDoc Label="AssociatedComputationRing:matrix">
## <ManSection>
## <Oper Arg="mat" Name="AssociatedComputationRing" Label="for homalg local matrices"/>
## <Returns>a &homalg; ring</Returns>
## <Description>
## Internally there is a ring, in which computations take place. This is either the global ring or a (not fully working) external pre ring in &Singular; with Mora's algorithm.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AssociatedComputationRing,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep ],
function( A )
return AssociatedComputationRing( HomalgRing(A) );
end
);
## <#GAPDoc Label="AssociatedGlobalRing:ring">
## <ManSection>
## <Oper Arg="R" Name="AssociatedGlobalRing" Label="for homalg local rings"/>
## <Returns>a (global) &homalg; ring</Returns>
## <Description>
## The global &homalg; ring, from which the local ring <A>R</A> was created.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AssociatedGlobalRing,
"for homalg local rings",
[ IsHomalgLocalRingRep ],
function( R )
if IsBound( R!.AssociatedGlobalRing ) then
return R!.AssociatedGlobalRing;
else
return R!.ring;
fi;
end
);
## <#GAPDoc Label="AssociatedGlobalRing:element">
## <ManSection>
## <Oper Arg="r" Name="AssociatedGlobalRing" Label="for homalg local ring elements"/>
## <Returns>a (global) &homalg; ring</Returns>
## <Description>
## The global &homalg; ring, from which the local ring element <A>r</A> was created.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AssociatedGlobalRing,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( r )
return AssociatedGlobalRing( HomalgRing( r ) );
end
);
## <#GAPDoc Label="AssociatedGlobalRing:matrix">
## <ManSection>
## <Oper Arg="mat" Name="AssociatedGlobalRing" Label="for homalg local matrices"/>
## <Returns>a (global) &homalg; ring</Returns>
## <Description>
## The global &homalg; ring, from which the local matrix <A>mat</A> was created.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AssociatedGlobalRing,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep ],
function( A )
return AssociatedGlobalRing( HomalgRing(A) );
end
);
## <#GAPDoc Label="Numerator:element">
## <ManSection>
## <Oper Arg="r" Name="Numerator" Label="for homalg local ring elements"/>
## <Returns>a (global) &homalg; ring element</Returns>
## <Description>
## The numerator from a local ring element <A>r</A>, which is a &homalg; ring element from the computation ring.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Numerator,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( r )
return r!.numer;
end
);
## <#GAPDoc Label="Denominator:element">
## <ManSection>
## <Oper Arg="r" Name="Denominator" Label="for homalg local ring elements"/>
## <Returns>a (global) &homalg; ring element</Returns>
## <Description>
## The denominator from a local ring element <A>r</A>, which is a &homalg; ring element from the computation ring.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Denominator,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( r )
return r!.denom;
end
);
## <#GAPDoc Label="Numerator:matrix">
## <ManSection>
## <Oper Arg="mat" Name="Numerator" Label="for homalg local matrices"/>
## <Returns>a (global) &homalg; matrix</Returns>
## <Description>
## The numerator from a local matrix <A>mat</A>, which is a &homalg; matrix from the computation ring.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Numerator,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep ],
function( M )
return Eval( M )[1];
end
);
## <#GAPDoc Label="Denominator:matrix">
## <ManSection>
## <Oper Arg="mat" Name="Denominator" Label="for homalg local matrices"/>
## <Returns>a (global) &homalg; ring element</Returns>
## <Description>
## The denominator from a local matrix <A>mat</A>, which is a &homalg; matrix from the computation ring.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Denominator,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep ],
function( M )
return Eval( M )[2];
end
);
## <#GAPDoc Label="Name">
## <ManSection>
## <Oper Arg="r" Name="Name" Label="for homalg local ring elements"/>
## <Returns>a string</Returns>
## <Description>
## The name of the local ring element <A>r</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Name,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( o )
local name, nnumer, ndenom, s1, s2, s3, s4;
if IsHomalgInternalRingRep( AssociatedComputationRing( o ) ) then
name := String;
else
name := Name;
fi;
nnumer := name( Numerator( o ) );
ndenom := name( Denominator( o ) );
if Length( nnumer ) <= 2 or not ( '+' in nnumer or '-' in nnumer ) then
s1 := "";
s2 := "";
else
s1 := "(";
s2 := ")";
fi;
if Length( ndenom ) <= 2 or not ( '+' in ndenom or '-' in ndenom ) then
s3 := "";
s4 := "";
else
s3 := "(";
s4 := ")";
fi;
return Flat( [ s1, nnumer, s2, "/", s3, ndenom, s4 ] );
end
);
InstallMethod( String,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( o )
return Name( o );
end
);
##
InstallMethod( String,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( r )
if IsOne( Denominator( r ) ) then
return String( Numerator( r ) );
fi;
TryNextMethod( );
end );
##
InstallMethod( BlindlyCopyMatrixPropertiesToLocalMatrix, ## under construction
"for homalg matrices",
[ IsHomalgMatrix, IsHomalgLocalMatrixRep ],
function( S, T )
local c;
for c in [ NumberRows, NumberColumns ] do
if Tester( c )( S ) then
Setter( c )( T, c( S ) );
fi;
od;
for c in [ IsZero, IsOne, IsDiagonalMatrix ] do
if Tester( c )( S ) and c( S ) then
Setter( c )( T, c( S ) );
fi;
od;
end
);
## <#GAPDoc Label="SetMatElm">
## <ManSection>
## <Oper Arg="mat, i, j, r, R" Name="SetMatElm" Label="for homalg local matrices"/>
## <Description>
## Changes the entry (<A>i,j</A>) of the local matrix <A>mat</A> to the value <A>r</A>. Here <A>R</A> is the (local) &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( SetMatElm,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep and IsMutable, IsPosInt, IsPosInt, IsHomalgLocalRingElementRep, IsHomalgLocalRingRep ],
function( M, r, c, s, R )
local m, cR, N, M2, e;
cR := AssociatedComputationRing( R );
#Create a new matrix with only the one entry
N := HomalgInitialMatrix( NumberRows( M ), NumberColumns( M ), cR );
N[ r, c ] := Numerator( s );
ResetFilterObj( N, IsInitialMatrix );
N := HomalgLocalMatrix( N, Denominator( s ), R );
#set the corresponding entry to zero in the other matrix
m := Eval( M );
M2 := m[1];
M2[ r, c ] := Zero( cR );
#add these matrices
e := Eval( HomalgLocalMatrix( M2, m[2], R ) + N );
SetIsMutableMatrix( e[1], true );
M!.Eval := e;
end
);
## <#GAPDoc Label="AddToMatElm">
## <ManSection>
## <Oper Arg="mat, i, j, r, R" Name="AddToMatElm" Label="for homalg local matrices"/>
## <Description>
## Changes the entry (<A>i,j</A>) of the local matrix <A>mat</A> by adding the value <A>r</A> to it. Here <A>R</A> is the (local) &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( AddToMatElm,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep and IsMutable, IsPosInt, IsPosInt, IsHomalgLocalRingElementRep, IsHomalgLocalRingRep ],
function( M, r, c, s, R )
local N, e;
#create a matrix with just one entry (i,j), which is s
N := HomalgInitialMatrix( NumberRows( M ), NumberColumns( M ), AssociatedComputationRing( R ) );
N[ r, c ] := Numerator( s );
ResetFilterObj( N, IsInitialIdentityMatrix );
N := HomalgLocalMatrix( N, Denominator( s ), R );
#and add this matrix to M
e := Eval( M + N );
SetIsMutableMatrix( e[1], true );
M!.Eval := e;
end
);
## <#GAPDoc Label="MatElmAsString">
## <ManSection>
## <Oper Arg="mat, i, j, R" Name="MatElmAsString" Label="for homalg local matrices"/>
## <Returns>a string</Returns>
## <Description>
## Returns the entry (<A>i,j</A>) of the local matrix <A>mat</A> as a string. Here <A>R</A> is the (local) &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( MatElmAsString,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep, IsPosInt, IsPosInt, IsHomalgLocalRingRep ],
function( M, r, c, R )
local m;
m := Eval( M );
return Concatenation( [ "(", MatElmAsString( m[1], r, c, AssociatedComputationRing( R ) ), ")/(", Name( m[2] ), ")" ] );
end
);
## <#GAPDoc Label="MatElm">
## <ManSection>
## <Oper Arg="mat, i, j, R" Name="MatElm" Label="for homalg local matrices"/>
## <Returns>a local ring element</Returns>
## <Description>
## Returns the entry (<A>i,j</A>) of the local matrix <A>mat</A>. Here <A>R</A> is the (local) &homalg; ring involved in these computations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( MatElm,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep, IsPosInt, IsPosInt, IsHomalgLocalRingRep ],
function( M, r, c, R )
local m;
m :=Eval( M );
return HomalgLocalRingElement( MatElm( m[1], r, c, AssociatedComputationRing( R ) ), m[2], R );
end
);
## <#GAPDoc Label="Cancel">
## <ManSection>
## <Oper Arg="a, b" Name="Cancel" Label="for pairs of ring elements"/>
## <Returns>two ring elements</Returns>
## <Description>
## For <M><A>a</A>=a'*c</M> and <M><A>b</A>=b'*c</M> return <M>a'</M> and <M>b'</M>. The exact form of <M>c</M> depends on whether a procedure for gcd computation is included in the ring package.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Cancel,
"for pairs of ring elements from the computation ring",
[ IsRingElement, IsRingElement ],
function( a, b )
local za, zb, z, ma, mb, o, g;
za := IsZero( a );
zb := IsZero( b );
if za and zb then
z := Zero( a );
return [ z, z ];
elif za then
return [ Zero( a ), One( a ) ];
elif zb then
return [ One( a ), Zero( a ) ];
fi;
if IsOne( a ) then
return [ One( a ), b ];
elif IsOne( b ) then
return [ a, One( a ) ];
fi;
ma := a = -One( a );
mb := b = -One( b );
if ma and mb then
o := One( a );
return [ o, o ];
elif ma then
return [ One( a ), -b ];
elif mb then
return [ -a, One( b ) ];
fi;
g := Gcd( a, b );
return [ a / g, b / g ];
end );
InstallMethod( Cancel,
"for pairs of ring elements from the computation ring",
[ IsHomalgRingElement, IsHomalgRingElement ],
function( a, b )
local R, za, zb, z, ma, mb, o, RP, result;
R := HomalgRing( a );
if R = fail then
TryNextMethod( );
elif not HasRingElementConstructor( R ) then
Error( "no ring element constructor found in the ring\n" );
fi;
za := IsZero( a );
zb := IsZero( b );
if za and zb then
z := Zero( R );
return [ z, z ];
elif za then
return [ Zero( R ), One( R ) ];
elif zb then
return [ One( R ), Zero( R ) ];
fi;
if IsOne( a ) then
return [ One( R ), b ];
elif IsOne( b ) then
return [ a, One( R ) ];
fi;
ma := IsMinusOne( a );
mb := IsMinusOne( b );
if ma and mb then
o := One( R );
return [ o, o ];
elif ma then
return [ One( R ), -b ];
elif mb then
return [ -a, One( R ) ];
fi;
RP := homalgTable( R );
if IsBound(RP!.CancelGcd) then
result := RP!.CancelGcd( a, b );
result := List( result, x -> HomalgRingElement( x, R ) );
Assert( 6, result[1] * b = result[2] * a );
return result;
else #fallback: no cancelation
return [ a, b ];
fi;
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement and IsMinusOne, IsRingElement ],
function( a, b )
return [ One( b ), -b ];
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement, IsRingElement and IsMinusOne ],
function( a, b )
return [ -a, One( a ) ];
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement and IsMinusOne, IsRingElement and IsMinusOne ],
function( a, b )
local o;
o := One( a );
return [ o, o ];
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement and IsOne, IsRingElement ],
function( a, b )
return [ One( a ), b ];
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement, IsRingElement and IsOne ],
function( a, b )
return [ a, One( b ) ];
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement and IsZero, IsRingElement ],
function( a, b )
return [ Zero( a ), One( a ) ];
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement, IsRingElement and IsZero ],
function( a, b )
return [ One( b ), Zero( b ) ];
end );
##
InstallMethod( Cancel,
"for pairs of global ring elements",
[ IsRingElement and IsZero, IsRingElement and IsZero ],
function( a, b )
local z;
z := Zero( a );
return [ z, z ];
end );
##
InstallMethod( SaveHomalgMatrixToFile,
"for local rings",
[ IsString, IsHomalgMatrix, IsHomalgLocalRingRep ],
function( filename, M, R )
local ComputationRing, NumerString, DenomString;
if LoadPackage( "HomalgToCAS" ) <> true then
Error( "the package HomalgToCAS failed to load\n" );
fi;
NumerString := Concatenation( filename, "_numerator" );
DenomString := Concatenation( filename, "_denominator" );
ComputationRing := AssociatedComputationRing( M );
SaveHomalgMatrixToFile( NumerString, Numerator( M ), ComputationRing );
SaveHomalgMatrixToFile( DenomString, HomalgMatrix( [ Denominator( M ) ], 1, 1, ComputationRing ), ComputationRing );
return [ NumerString, DenomString ];
end );
##
InstallMethod( LoadHomalgMatrixFromFile,
"for local rings",
[ IsString, IsInt, IsInt, IsHomalgLocalRingRep ],
function( filename, r, c, R )
local ComputationRing, numer, denom, homalgIO;
ComputationRing := AssociatedComputationRing( R );
if IsExistingFile( Concatenation( filename, "_numerator" ) ) and IsExistingFile( Concatenation( filename, "_denominator" ) ) then
numer := LoadHomalgMatrixFromFile( Concatenation( filename, "_numerator" ), r, c, ComputationRing );
denom := LoadHomalgMatrixFromFile( Concatenation( filename, "_denominator" ), r, c, ComputationRing );
denom := denom[ 1, 1 ];
elif IsExistingFile( filename ) then
numer := LoadHomalgMatrixFromFile( filename, r, c, ComputationRing );
denom := One( ComputationRing );
else
Error( "file does not exist" );
fi;
return HomalgLocalMatrix( numer, denom, R );
end );
##
InstallMethod( CreateHomalgMatrixFromString,
"constructor for homalg matrices over local rings",
[ IsString, IsInt, IsInt, IsHomalgLocalRingRep ],
function( s, r, c, R )
local mat;
mat := CreateHomalgMatrixFromString( s, r, c, AssociatedComputationRing( R ) );
return R * mat;
end );
##
InstallMethod( SetRingProperties,
"for homalg local rings",
[ IsHomalgRing and IsLocal ],
function( S )
local R;
R := S!.AssociatedGlobalRing;
# SetCoefficientsRing( S, R );
# SetCharacteristic( S, Characteristic( R ) );
if HasIsCommutative( R ) and IsCommutative( R ) then
SetIsCommutative( S, true );
fi;
# if HasGlobalDimension( R ) then
# SetGlobalDimension( S, GlobalDimension( R ) ); ## would be wrong
# fi;
# if HasKrullDimension( R ) then
# SetKrullDimension( S, KrullDimension( R ) ); ## would be wrong
# fi;
#
# SetIsIntegralDomain( S, true ); ## would be wrong, see Hartshorne
if HasCoefficientsRing( R ) then
SetCoefficientsRing( S, CoefficientsRing( R ) );
fi;
if HasIndeterminatesOfPolynomialRing( R ) then
SetIndeterminatesOfPolynomialRing( S,
List( IndeterminatesOfPolynomialRing( R ),
x -> x / S )
);
fi;
if HasIsFreePolynomialRing( R ) and IsFreePolynomialRing( R ) then
SetIsIntegralDomain( S, true );
fi;
if HasKrullDimension( R ) and HasIsIntegralDomain( R ) and IsIntegralDomain( R ) then
SetKrullDimension( S, KrullDimension( R ) );
fi;
# SetBasisAlgorithmRespectsPrincipalIdeals( S, true ); ## ???
end );
####################################
#
# constructor functions and methods:
#
####################################
## <#GAPDoc Label="LocalizeAt">
## <ManSection>
## <Oper Arg="R, l" Name="LocalizeAt" Label= "for a commutative ring and a maximal ideal"/>
## <Returns>a local ring</Returns>
## <Description>
## If <A>l</A> is a list of elements of the global ring <A>R</A> generating a maximal ideal, the method creates the corresponding localization of <A>R</A> at the complement of the maximal ideal.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( LocalizeAt,
"constructor for homalg localized rings",
[ IsHomalgRing and IsCommutative, IsList ],
function( globalR, ideal_gens )
local RP, localR, n_gens, gens;
if not IsString( ideal_gens ) then
ideal_gens := List( ideal_gens,
function( x )
if not IsString( x ) then
return String( x );
fi;
return x;
end );
ideal_gens := JoinStringsWithSeparator( ideal_gens );
fi;
ideal_gens := ParseListOfIndeterminates( SplitString( ideal_gens, "," ) );
if ForAny( ideal_gens, IsString ) then
ideal_gens :=
List( ideal_gens,
function( x )
if IsString( x ) then
return x / globalR;
fi;
return x;
end );
fi;
RP := CreateHomalgTableForLocalizedRings( globalR );
if LoadPackage( "RingsForHomalg" ) <> true then
Error( "the package RingsForHomalg failed to load\n" );
fi;
if ValueGlobal( "IsHomalgExternalRingInSingularRep" )( globalR ) then
UpdateMacrosOfLaunchedCAS( LocalizeRingMacrosForSingular, homalgStream( globalR ) );
AppendToAhomalgTable( RP, HomalgTableForLocalizedRingsForSingularTools );
fi;
## create the local ring
localR := CreateHomalgRing( globalR, [ TheTypeHomalgLocalRing, TheTypeHomalgLocalMatrix ], HomalgLocalRingElement, RP );
SetConstructorForHomalgMatrices( localR,
function( arg )
local R, r, ar, M;
R := arg[Length( arg )];
#at least be able to construct 1x1-matrices from lists of ring elements for the fallback IsUnit
if IsList( arg[1] ) and Length( arg[1] ) = 1 and IsHomalgLocalRingElementRep( arg[1][1] ) then
r := arg[1][1];
return HomalgLocalMatrix( HomalgMatrix( [ Numerator( r ) ], 1, 1, AssociatedComputationRing( R ) ), Denominator( r ), R );
fi;
ar := List( arg,
function( i )
if IsHomalgLocalRingRep( i ) then
return AssociatedComputationRing( i );
else
return i;
fi;
end );
M := CallFuncList( HomalgMatrix, ar );
return HomalgLocalMatrix( M, R );
end );
## for the view methods:
## <A homalg local ring>
## <A matrix over a local ring>
localR!.description := " local";
SetIsLocal( localR, true );
#Set the ideal, at which we localize
n_gens := Length( ideal_gens );
gens := HomalgMatrix( ideal_gens, n_gens, 1, globalR );
SetGeneratorsOfMaximalLeftIdeal( localR, gens );
gens := HomalgMatrix( ideal_gens, 1, n_gens, globalR );
SetGeneratorsOfMaximalRightIdeal( localR, gens );
localR!.AssociatedGlobalRing := globalR;
localR!.AssociatedComputationRing := globalR;
SetRingProperties( localR );
return localR;
end );
## <#GAPDoc Label="LocalizeAtZero">
## <ManSection>
## <Oper Arg="R" Name="LocalizeAtZero" Label= "for a free polynomial ring"/>
## <Returns>a local ring</Returns>
## <Description>
## This method creates the corresponding localization of <A>R</A> at the complement of the maximal ideal generated by the indeterminates ("at zero").
## </Description>
## </ManSection>
## <#/GAPDoc>
InstallMethod( LocalizeAtZero,
"constructor for homalg localized rings",
[ IsHomalgRing and IsFreePolynomialRing ],
function( globalR )
return LocalizeAt( globalR, IndeterminatesOfPolynomialRing( globalR ) );
end );
## <#GAPDoc Label="HomalgLocalRingElement">
## <ManSection>
## <Func Arg="numer, denom, R" Name="HomalgLocalRingElement" Label="constructor for local ring elements using numerator and denominator"/>
## <Func Arg="numer, R" Name="HomalgLocalRingElement" Label="constructor for local ring elements using a given numerator and one as denominator"/>
## <Returns>a local ring element</Returns>
## <Description>
## Creates the local ring element <M><A>numer</A>/<A>denom</A></M> or in the second case <M><A>numer</A>/1</M> for the local ring <A>R</A>. Both <A>numer</A> and <A>denom</A> may either be a string describing a valid global ring element or from the global ring or computation ring.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallGlobalFunction( HomalgLocalRingElement,
function( arg )
local nargs, numer, ring, ar, properties, denom, computationring, c, r;
nargs := Length( arg );
if nargs = 0 then
Error( "empty input\n" );
fi;
numer := arg[1];
if IsHomalgLocalRingElementRep( numer ) then
##a local ring element as first argument will just be returned
return numer;
elif nargs = 2 then
## extract the properties of the global ring element
if IsHomalgRing( arg[2] ) then
ring := arg[2];
ar := [ numer, One( AssociatedComputationRing( ring ) ), ring ];
properties := KnownTruePropertiesOfObject( numer ); ## FIXME: a huge potential for problems
Add( ar, List( properties, ValueGlobal ) ); ## at least an empty list is inserted; avoids infinite loops
return CallFuncList( HomalgLocalRingElement, ar );
fi;
fi;
properties := [ ];
for ar in arg{[ 2 .. nargs ]} do
if not IsBound( ring ) and IsHomalgRing( ar ) then
ring := ar;
elif IsList( ar ) and ForAll( ar, IsFilter ) then
Append( properties, ar );
elif not IsBound( denom ) and ( IsRingElement( ar ) or IsString( ar ) ) then
denom := ar;
else
Error( "this argument (now assigned to ar) should be in { IsHomalgRing, IsList( IsFilter ), IsRingElement, IsString }\n" );
fi;
od;
computationring := AssociatedComputationRing( ring );
if not IsBound( denom ) then
denom := One( numer );
fi;
if not IsHomalgRingElement( numer ) then
numer := HomalgRingElement( numer, computationring );
fi;
if not IsHomalgRingElement( denom ) then
denom := HomalgRingElement( denom, computationring );
fi;
c := Cancel( numer, denom );
numer := c[1] / computationring;
denom := c[2] / computationring;
if IsBound( ring ) then
r := rec( numer := numer, denom := denom, ring := ring );
## Objectify:
Objectify( TheTypeHomalgLocalRingElement, r );
fi;
if properties <> [ ] then
for ar in properties do
Setter( ar )( r, true );
od;
fi;
return r;
end );
## <#GAPDoc Label="HomalgLocalMatrix">
## <ManSection>
## <Func Arg="numer, denom, R" Name="HomalgLocalMatrix" Label="constructor for local matrices using numerator and denominator"/>
## <Func Arg="numer, R" Name="HomalgLocalMatrix" Label="constructor for local matrices using a given numerator and one as denominator"/>
## <Returns>a local matrix</Returns>
## <Description>
## Creates the local matrix <M><A>numer</A>/<A>denom</A></M> or in the second case <M><A>numer</A>/1</M> for the local ring <A>R</A>. Both <A>numer</A> and <A>denom</A> may either be from the global ring or the computation ring.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( HomalgLocalMatrix,
"constructor for matrices over localized rings",
[ IsHomalgMatrix, IsRingElement, IsHomalgLocalRingRep ],
function( A, r, R )
local G, type, matrix, HookDenom, ComputationRing, rr, AA;
G := HomalgRing( A );
ComputationRing := AssociatedComputationRing( R );
if IsBound( A!.Denominator ) then
HookDenom := A!.Denominator;
Unbind( A!.Denominator );
else
HookDenom := One( ComputationRing );
fi;
if not ( IsHomalgRingElement( r ) and IsIdenticalObj( ComputationRing , HomalgRing( r ) ) ) then
rr := r / ComputationRing;
else
rr := r;
fi;
if not IsIdenticalObj( ComputationRing , HomalgRing( A ) ) then
AA := ComputationRing * A;
else
AA := A;
fi;
matrix := rec( ring := R );
ObjectifyWithAttributes(
matrix, TheTypeHomalgLocalMatrix,
Eval, [ AA, rr * HookDenom ]
);
BlindlyCopyMatrixPropertiesToLocalMatrix( A, matrix );
return matrix;
end );
##
InstallMethod( HomalgLocalMatrix,
"constructor for matrices over localized rings",
[ IsHomalgMatrix, IsHomalgLocalRingRep ],
function( A, R )
return HomalgLocalMatrix( A, One( AssociatedComputationRing( R ) ), R );
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgLocalRingRep, IsHomalgMatrix ],
function( R, m )
if IsHomalgLocalMatrixRep( m ) then
TryNextMethod( );
fi;
return HomalgLocalMatrix( AssociatedComputationRing( R ) * m, R );
end );
##
InstallMethod( \*,
"for matrices over local rings",
[ IsHomalgRing, IsHomalgLocalMatrixRep ],
function( R, m )
if not IsOne( Denominator( m ) ) then
TryNextMethod( );
fi;
return R * Numerator( m );
end );
##
InstallMethod( PostMakeImmutable,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep and HasEval ],
function( A )
MakeImmutable( Numerator( A ) );
end );
##
InstallMethod( SetIsMutableMatrix,
"for homalg local matrices",
[ IsHomalgLocalMatrixRep, IsBool ],
function( A, b )
if b = true then
SetFilterObj( A, IsMutable );
else
ResetFilterObj( A, IsMutable );
fi;
SetIsMutableMatrix( Numerator( A ), b );
end );
##
InstallMethod( RingOfDerivations,
"for local rings",
[ IsHomalgLocalRingRep ],
function( Rm )
local R;
R := AssociatedGlobalRing( Rm );
return RingOfDerivations( R );
end );
####################################
#
# View, Print, and Display methods:
#
####################################
##
InstallMethod( Display,
"for homalg local ring elements",
[ IsHomalgLocalRingElementRep ],
function( r )
Print( Name( r ), "\n" );
end );
##
InstallMethod( Display,
"for homalg matrices over a homalg local ring",
[ IsHomalgLocalMatrixRep ],
function( A )
local a;
a := Eval( A );
Display( a[1] );
if IsHomalgInternalRingRep( AssociatedComputationRing( A ) ) then
Print( "/ ", a[2], "\n" );
else
if IsOne( a[2] ) or IsMinusOne( a[2] ) then
Print( "/ ", Name( a[2] ), "\n" );
else
Print( "/(", Name( a[2] ), ")\n" );
fi;
fi;
end );
[ Dauer der Verarbeitung: 0.5 Sekunden
(vorverarbeitet)
]
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2026-04-02
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