|
# SPDX-License-Identifier: GPL-2.0-or-later
# Modules: A homalg based package for the Abelian category of finitely presented modules over computable rings
#
# Implementations
#
## LIMAP = Logical Implications for homalg ring MAPs
####################################
#
# representations:
#
####################################
####################################
#
# methods for properties:
#
####################################
##
InstallMethod( IsMonomorphism,
"LIMAP: for homalg ring maps",
[ IsHomalgRingMapRep ],
function( phi )
return IsMorphism( phi ) and IsZero( Kernel( phi ) );
end );
####################################
#
# methods for attributes:
#
####################################
##
InstallMethod( KernelSubobject,
"for homalg ring maps",
[ IsHomalgRingMap ],
function( phi )
local rel, S;
rel := GeneratorsOfKernelOfRingMap( phi );
S := HomalgRing( rel );
if IsBound( phi!.left ) and phi!.left = false then
S := S * 1;
else
S := 1 * S; ## the default
fi;
return Subobject( rel, S );
end );
##
InstallMethod( KernelEmb,
"for homalg ring maps",
[ IsHomalgRingMap ],
function( phi )
return EmbeddingInSuperObject( KernelSubobject( phi ) );
end );
##
InstallMethod( IdealContainedInKernelViaEliminateOverBaseRing,
[ IsHomalgRingMap ],
function( phi )
local T, mat, indetsT, lT, S, indetsS, lS, imgs, elim, zero, indets, iota;
T := Range( phi );
mat := MatrixOfRelations( T );
T := AmbientRing( T );
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
Error( imgs, " is not a sublist of ", indetsT, "\n" );
fi;
mat := EliminateOverBaseRing( 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 );
if IsBound( phi!.left ) and phi!.left = false then
S := S * 1;
else
S := 1 * S; ## the default
fi;
return Subobject( mat, S );
end );
####################################
#
# methods for operations:
#
####################################
## <#GAPDoc Label="Kernel:ringmap">
## <ManSection>
## <Meth Arg="phi" Name="Kernel" Label="for ring maps"/>
## <Returns>a &homalg; module</Returns>
## <Description>
## The kernel ideal of the ring map <A>phi</A> as an abstract module.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( Kernel,
"for homalg ring maps",
[ IsHomalgRingMap ],
function( phi )
return UnderlyingObject( KernelSubobject( phi ) );
end );
[ zur Elbe Produktseite wechseln0.25Quellennavigators
Analyse erneut starten
]
|
