|
# SPDX-License-Identifier: GPL-2.0-or-later
# MatricesForHomalg: Matrices for the homalg project
#
# Implementations
#
## <#GAPDoc Label="RingMaps:intro">
## A &homalg; ring map is a data structure for maps between finitely generated rings. &homalg; more or less provides the basic
## declarations and installs the generic methods for ring maps, but it is up to other high level packages to install methods
## applicable to specific rings. For example, the package &Sheaves; provides methods for ring maps of (finitely generated) affine rings.
## <#/GAPDoc>
####################################
#
# representations:
#
####################################
## <#GAPDoc Label="IsHomalgRingMapRep">
## <ManSection>
## <Filt Type="Representation" Arg="phi" Name="IsHomalgRingMapRep"/>
## <Returns><C>true</C> or <C>false</C></Returns>
## <Description>
## The &GAP; representation of &homalg; ring maps. <P/>
## (It is a representation of the &GAP; category <Ref Filt="IsHomalgRingMap"/>.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsHomalgRingMapRep",
IsHomalgRingMap,
[ ] );
# a new family:
BindGlobal( "TheFamilyOfHomalgRingMaps",
NewFamily( "TheFamilyOfHomalgRingMaps" ) );
# two new types:
BindGlobal( "TheTypeHomalgRingMap",
NewType( TheFamilyOfHomalgRingMaps,
IsHomalgRingMapRep ) );
BindGlobal( "TheTypeHomalgRingSelfMap",
NewType( TheFamilyOfHomalgRingMaps,
IsHomalgRingMapRep and IsHomalgRingSelfMap ) );
####################################
#
# methods for operations:
#
####################################
##
InstallMethod( ImagesOfRingMap,
"for homalg ring maps",
[ IsHomalgRingMap ],
function( phi )
return EntriesOfHomalgMatrix( ImagesOfRingMapAsColumnMatrix( phi ) );
end );
##
InstallMethod( PreCompose,
"for two homalg ring maps",
[ IsHomalgRingMap, IsHomalgRingMap ],
function( psi, phi )
local S, T, images_psi, chi;
S := Source( psi );
T := Range( phi );
images_psi := ImagesOfRingMapAsColumnMatrix( psi );
chi := RingMap( Pullback( phi, images_psi ), S, T );
if HasIsIsomorphism( psi ) and IsIsomorphism( psi ) and
HasIsIsomorphism( phi ) and IsIsomorphism( phi ) then
SetIsIsomorphism( chi, true );
elif HasIsEpimorphism( psi ) and IsEpimorphism( psi ) and
HasIsEpimorphism( phi ) and IsEpimorphism( phi ) then
SetIsEpimorphism( chi, true );
elif HasIsMonomorphism( psi ) and IsMonomorphism( psi ) and
HasIsMonomorphism( phi ) and IsMonomorphism( phi ) then
SetIsMonomorphism( chi, true );
elif HasIsMorphism( psi ) and IsMorphism( psi ) and
HasIsMorphism( phi ) and IsMorphism( phi ) then
SetIsMorphism( chi, true );
fi;
return chi;
end );
####################################
#
# constructor functions and methods:
#
####################################
## <#GAPDoc Label="RingMap">
## <ManSection>
## <Oper Arg="images, S, T" Name="RingMap" Label="constructor for ring maps"/>
## <Returns>a &homalg; ring map</Returns>
## <Description>
## This constructor returns a ring map (homomorphism) of finitely generated rings/algebras. It is represented by the
## images <A>images</A> of the set of generators of the source &homalg; ring <A>S</A> in terms of the
## generators of the target ring <A>T</A> (&see; <Ref Sect="Rings:Constructors"/>). Unless the source ring is free
## <E>and</E> given on free ring/algebra generators the returned map will cautiously be indicated using
## parenthesis: <Q>homomorphism</Q>.
## If source and target are identical objects, and only then, the ring map is created as a selfmap.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InstallMethod( RingMap,
"for a homalg matrix and two homalg rings",
[ IsHomalgMatrix, IsHomalgRing, IsHomalgRing ],
function( images, S, T )
local c, map, type;
c := NumberColumns( images );
if not ( c = 1 or NumberRows( images ) = 1 ) then
Error( "the matrix must either has one row or one column\n" );
fi;
if c > 1 then
images := Involution( images );
fi;
if not IsIdenticalObj( T, HomalgRing( images ) ) then
images := T * images;
fi;
map := rec( );
if IsIdenticalObj( S, T ) then
type := TheTypeHomalgRingSelfMap;
else
type := TheTypeHomalgRingMap;
fi;
## Objectify:
ObjectifyWithAttributes(
map, type,
ImagesOfRingMapAsColumnMatrix, images,
Source, S,
Range, T );
return map;
end );
##
InstallMethod( RingMap,
"for a list and two homalg rings",
[ IsList, IsHomalgRing, IsHomalgRing ],
function( images, S, T )
local matrix, map;
matrix := HomalgMatrix( images, Length( images ), 1, T );
map := RingMap( matrix, S, T );
SetImagesOfRingMap( map, images );
return map;
end );
##
InstallMethod( RingMap,
"for a homalg matrix and a homalg ring",
[ IsHomalgMatrix, IsHomalgRing ],
function( mat, R )
local r, indets;
r := NumberRows( mat );
if NumberColumns( mat ) <> r then
Error( "the matrix is not quadratic\n" );
fi;
indets := Indeterminates( R );
if Length( indets ) <> r then
Error( "the number of indeterminates does not match the number of rows of the matrix\n" );
fi;
indets := HomalgMatrix( indets, r, 1, R );
if not IsIdenticalObj( R, HomalgRing( mat ) ) then
mat := R * mat;
fi;
return RingMap( mat * indets, R, R );
end );
##
InstallMethod( RingMap,
"for a homalg matrix",
[ IsHomalgMatrix ],
function( mat )
return RingMap( mat, HomalgRing( mat ) );
end );
##
InstallMethod( RingMap,
"for a homalg ring",
[ IsHomalgRing ],
function( R )
local map;
if IsBound( R!.IdentityMorphism ) then
return R!.IdentityMorphism;
fi;
map := RingMap( Indeterminates( R ), R, R );
SetIsMorphism( map, true );
SetIsIsomorphism( map, true );
SetIsOne( map, true );
R!.IdentityMorphism := map;
return map;
end );
####################################
#
# View, Print, and Display methods:
#
####################################
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingMap ],
function( o )
Print( "<A" );
if HasIsMorphism( o ) then
if IsMorphism( o ) then
Print( " homomorphism of" );
else
Print( " non-well-defined map of" );
fi;
else
Print( " \"homomorphism\" of" );
fi;
Print( " rings>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingMap and IsMonomorphism ], 996,
function( o )
Print( "<A monomorphism of rings>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingMap and IsEpimorphism ], 997,
function( o )
Print( "<An epimorphism of rings>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingMap and IsIsomorphism ], 1000,
function( o )
Print( "<An isomorphism of rings>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingSelfMap ],
function( o )
Print( "<A" );
if HasIsZero( o ) then ## if this method applies and HasIsZero is set we already know that o is a non-zero map of homalg rings
Print( " non-zero" );
elif not ( HasIsMorphism( o ) and not IsMorphism( o ) ) then
Print( "n" );
fi;
if HasIsMorphism( o ) then
if IsMorphism( o ) then
Print( " endomorphism of" );
else
Print( " non-well-defined self-map of" );
fi;
else
Print( " endo\"morphism\" of" );
fi;
Print( " rings>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingSelfMap and IsMonomorphism ],
function( o )
Print( "<A monic endomorphism of a ring>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingSelfMap and IsEpimorphism ], 996,
function( o )
Print( "<An epic endomorphism a ring>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingSelfMap and IsAutomorphism ], 999,
function( o )
Print( "<An automorphism of a ring>" );
end );
##
InstallMethod( ViewObj,
"for homalg ring maps",
[ IsHomalgRingSelfMap and IsIdentityMorphism ], 1000,
function( o )
Print( "<The identity morphism of a ring>" );
end );
##
InstallMethod( Display,
"for homalg ring maps",
[ IsHomalgRingMapRep ],
function( o )
ViewObj( Range( o ) );
Print( "\n\ ^\n |\n" );
ViewObj( ImagesOfRingMap( o ) );
Print( "\n |\n |\n" );
ViewObj( Source( o ) );
Print( "\n" );
end );
[ Dauer der Verarbeitung: 0.32 Sekunden
(vorverarbeitet)
]
|
