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# SPDX-License-Identifier: GPL-2.0-or-later
# MatricesForHomalg: Matrices for the homalg project
#
# Implementations
#
## LIMAP = Logical Implications for homalg ring MAPs
####################################
#
# global variables:
#
####################################
# a central place for configuration variables:
InstallValue( LIMAP,
rec(
color := "\033[4;30;46m",
)
);
##
InstallValue( LogicalImplicationsForHomalgRingMaps,
[
[ IsMonomorphism,
"implies", IsMorphism ],
[ IsEpimorphism,
"implies", IsMorphism ],
[ IsAutomorphism,
"implies", IsIsomorphism ],
[ IsEpimorphism, "and", IsMonomorphism,
"imply", IsIsomorphism ],
[ IsIdentityMorphism,
"implies", IsAutomorphism ],
] );
##
InstallValue( LogicalImplicationsForHomalgRingSelfMaps,
[
[ IsIsomorphism,
"implies", IsAutomorphism ],
] );
####################################
#
# logical implications methods:
#
####################################
InstallLogicalImplicationsForHomalgBasicObjects( LogicalImplicationsForHomalgRingMaps, IsHomalgRingMap );
InstallLogicalImplicationsForHomalgBasicObjects( LogicalImplicationsForHomalgRingSelfMaps, IsHomalgRingSelfMap );
####################################
#
# immediate methods for properties:
#
####################################
##
InstallImmediateMethod( IsAutomorphism,
IsHomalgRingMap, 0,
function( phi )
if not IsIdenticalObj( Source( phi ), Range( phi ) ) then
return false;
fi;
TryNextMethod( );
end );
####################################
#
# methods for properties:
#
####################################
##
InstallMethod( IsIsomorphism,
"LIMAP: for homalg morphisms",
[ IsHomalgRingMap ],
function( phi )
return IsEpimorphism( phi ) and IsMonomorphism( phi );
end );
##
InstallMethod( IsAutomorphism,
"LIMAP: for homalg ring maps",
[ IsHomalgRingMapRep ],
function( phi )
return IsHomalgRingSelfMap( phi ) and IsIsomorphism( phi );
end );
####################################
#
# methods for attributes:
#
####################################
##
InstallMethod( DataOfCoordinateRingOfGraph,
"LIMAP: for homalg ring maps",
[ IsHomalgRingMapRep ],
function( phi )
local S, T, indetsS, indetsT, r, relS, relT, ST, images, rel, G;
S := Source( phi );
T := Range( phi );
if HasIndeterminatesOfPolynomialRing( S ) then
indetsS := IndeterminatesOfPolynomialRing( S );
r := CoefficientsRing( S );
elif HasAmbientRing( S ) and HasIndeterminatesOfPolynomialRing( AmbientRing( S ) ) then
indetsS := IndeterminatesOfPolynomialRing( AmbientRing( S ) );
r := CoefficientsRing( AmbientRing( S ) );
else
indetsS := [ ];
r := S;
fi;
if HasIndeterminatesOfPolynomialRing( T ) then
indetsT := IndeterminatesOfPolynomialRing( T );
if not IsIdenticalObj( r, CoefficientsRing( T ) ) then
Error( "different coefficient rings are not supported yet\n" );
fi;
elif HasAmbientRing( T ) and HasIndeterminatesOfPolynomialRing( AmbientRing( T ) ) then
indetsT := IndeterminatesOfPolynomialRing( AmbientRing( T ) );
if not IsIdenticalObj( r, CoefficientsRing( AmbientRing( T ) ) ) then
Error( "different coefficient rings are not supported yet\n" );
fi;
fi;
if not IsBound( indetsT ) then
indetsT := [ ];
fi;
if not IsBound( indetsS ) then
indetsS := [ ];
fi;
if HasAmbientRing( S ) then
ST := PolynomialRing( AmbientRing( S ), List( indetsT, Name ) );
else
ST := PolynomialRing( S, List( indetsT, Name ) );
fi;
if HasRingRelations( S ) then
relS := ST * MatrixOfRelations( S );
if not NumberColumns( relS ) = 1 then
relS := Involution( relS );
fi;
fi;
if HasRingRelations( T ) then
relT := ST * MatrixOfRelations( T );
if not NumberColumns( relT ) = 1 then
relT := Involution( relT );
fi;
fi;
images := ImagesOfRingMap( phi );
images := List( images, x -> x / ST );
images := HomalgMatrix( images, Length( images ), 1, ST );
if not IsBound( relS ) then
relS := HomalgZeroMatrix( 0, 1, ST );
fi;
if not IsBound( relT ) then
relT := HomalgZeroMatrix( 0, 1, ST );
fi;
return [ [ ST, List( indetsS, y -> y / ST ), List( indetsT, x -> x / ST ) ],
[ relS, images, relT ] ];
end );
##
InstallMethod( CoordinateRingOfGraph,
"LIMAP: for homalg ring maps",
[ IsHomalgRingMapRep ],
function( phi )
local data, indetsS, indetsT, relS, relT, ST, images, rel, G;
data := DataOfCoordinateRingOfGraph( phi );
ST := data[1][1];
indetsS := data[1][2];
indetsT := data[1][3];
relS := data[2][1];
images := data[2][2];
relT := data[2][3];
rel := HomalgMatrix( indetsS, Length( indetsS ), 1, ST ) - images;
rel := UnionOfRows( relS, rel );
rel := UnionOfRows( rel, relT );
rel := HomalgRingRelationsAsGeneratorsOfLeftIdeal( rel );
G := ST / rel;
G!.indetsS := indetsS;
G!.indetsT := indetsT;
return G;
end );
##
InstallMethod( GeneratorsOfKernelOfRingMap,
"for homalg ring maps",
[ IsHomalgRingMap ],
function( phi )
local G, S, T, indetsS, indetsT, rel;
G := CoordinateRingOfGraph( phi );
S := Source( phi );
T := Range( phi );
indetsT := G!.indetsT;
rel := RingRelations( G );
rel := MatrixOfRelations( rel );
rel := Eliminate( rel, indetsT );
rel := S * rel;
rel := BasisOfRows( rel );
return rel;
end );
## KernelSubobject of projections
InstallMethod( GeneratorsOfKernelOfRingMap,
"for homalg ring maps",
[ IsHomalgRingMap ],
function( phi )
local T, mat, indetsT, lT, S, indetsS, lS, imgs, elim, zero, indets, iota;
T := Range( phi );
if HasRingRelations( T ) then
mat := MatrixOfRelations( T );
T := AmbientRing( T );
else
mat := HomalgZeroMatrix( 0, 1, T );
fi;
indetsT := Indeterminates( T );
lT := Length( indetsT );
S := Source( phi );
indetsS := Indeterminates( S );
lS := Length( indetsS );
imgs := ImagesOfRingMap( phi );
elim := Difference( indetsT, imgs );
if not Length( elim ) = lT - lS then
TryNextMethod( );
fi;
mat := Eliminate( mat, elim );
## computing a basis might be expensive (delaying the output)
## so leave it to the user or the next procedure
#mat := BasisOfRows( mat );
zero := Zero( S );
indets := ListWithIdenticalEntries( lT, zero );
imgs := PositionsProperty( indetsT, a -> a in imgs );
indets{imgs} := indetsS;
iota := RingMap( indets, T, S );
mat := Pullback( iota, mat );
mat := BasisOfRows( mat );
return mat;
end );
[ Dauer der Verarbeitung: 0.26 Sekunden
(vorverarbeitet)
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