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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 "fake" localized rings.
DeclareRepresentation( "IsHomalgFakeLocalRingRep",
IsHomalgRing
and IsHomalgRingOrFinitelyPresentedModuleRep,
[ "ring" ] );
DeclareRepresentation( "IsElementOfHomalgFakeLocalRingRep",
IsHomalgRingElement,
[ "pointer" ] );
DeclareRepresentation( "IsMatrixOverHomalgFakeLocalRingRep",
IsHomalgMatrix,
[ ] );
BindGlobal( "TheTypeHomalgFakeLocalRing",
NewType( TheFamilyOfHomalgRings,
IsHomalgFakeLocalRingRep ) );
BindGlobal( "TheTypeElementOfHomalgFakeLocalRing",
NewType( TheFamilyOfHomalgRingElements,
IsElementOfHomalgFakeLocalRingRep ) );
BindGlobal( "TheTypeMatrixOverHomalgFakeLocalRing",
NewType( TheFamilyOfHomalgMatrices,
IsMatrixOverHomalgFakeLocalRingRep ) );
InstallValue( CommonHomalgTableForLocalizedRingsAtPrimeIdeals,
rec(
RingName :=
function( R )
local globalR, baseR, name;
globalR := AssociatedGlobalRing( R );
if HasBaseRing( globalR ) then
baseR:= BaseRing( globalR );
else
baseR := CoefficientsRing( globalR );
fi;
if HasName( R ) then
return Name( R );
fi;
name := Concatenation( RingName( baseR ), "_< ", JoinStringsWithSeparator( EntriesOfHomalgMatrix( GeneratorsOfPrimeIdeal( R ) ), ", " ), " >" );
if HasIndeterminatesOfPolynomialRing( R ) then
name := Concatenation( "( ", name, " )", Concatenation( "[", JoinStringsWithSeparator( List( Indeterminates( R ), String ) ), "]" ) );
fi;
return name;
end,
)
);
####################################
#
# methods for operations:
#
####################################
##
InstallMethod( AssociatedComputationRing,
"for homalg fake local rings",
[ IsHomalgFakeLocalRingRep ],
function( R )
if not IsBound( R!.AssociatedComputationRing ) then
return R!.ring;
fi;
return R!.AssociatedComputationRing;
end );
##
InstallMethod( AssociatedComputationRing,
"for homalg fake local ring elements",
[ IsElementOfHomalgFakeLocalRingRep ],
function( r )
return AssociatedComputationRing( HomalgRing( r ) );
end );
##
InstallMethod( AssociatedComputationRing,
"for matrices over homalg fake local rings",
[ IsMatrixOverHomalgFakeLocalRingRep ],
function( A )
return AssociatedComputationRing( HomalgRing(A) );
end );
##
InstallMethod( AssociatedResidueClassRing,
"for homalg fake local rings",
[ IsHomalgFakeLocalRingRep ],
function( R )
return R!.AssociatedResidueClassRing;
end );
##
InstallMethod( AssociatedResidueClassRing,
"for homalg fake local rings",
[ IsElementOfHomalgFakeLocalRingRep ],
function( R )
return R!.AssociatedResidueClassRing;
end );
##
InstallMethod( AssociatedResidueClassRing,
"for homalg fake local rings",
[ IsMatrixOverHomalgFakeLocalRingRep ],
function( R )
return R!.AssociatedResidueClassRing;
end );
# ##
# InstallMethod( BaseRing,
# "for homalg fake local rings",
# [ IsHomalgFakeLocalRingRep ],
# function( R )
# return R!.BaseRing;
# end );
# ##
# InstallMethod( BaseRing,
# "for homalg fake local rings",
# [ IsElementOfHomalgFakeLocalRingRep ],
# function( R )
# return R!.BaseRing;
# end );
# ##
# InstallMethod( BaseRing,
# "for homalg fake local rings",
# [ IsMatrixOverHomalgFakeLocalRingRep ],
# function( R )
# return R!.BaseRing;
# end );
##
InstallMethod( AssociatedGlobalRing,
"for homalg fake local rings",
[ IsHomalgFakeLocalRingRep ],
function( R )
if not IsBound( R!.AssociatedGlobalRing ) then
return R!.ring;
fi;
return R!.AssociatedGlobalRing;
end );
##
InstallMethod( AssociatedGlobalRing,
"for homalg fake local ring elements",
[ IsElementOfHomalgFakeLocalRingRep ],
function( r )
return AssociatedGlobalRing( HomalgRing( r ) );
end );
##
InstallMethod( AssociatedGlobalRing,
"for homalg fake local matrices",
[ IsMatrixOverHomalgFakeLocalRingRep ],
function( A )
return AssociatedGlobalRing( HomalgRing( A ) );
end );
##
InstallMethod( Numerator,
"for homalg fake local matrices",
[ IsElementOfHomalgFakeLocalRingRep ],
function( p )
local R, r, coeffs;
R := AssociatedGlobalRing( p );
if IsZero( p ) then
return Zero( R );
fi;
if not IsBound( p!.Numerator ) then
if HasCoefficientsRing( R ) then
r := CoefficientsRing( R );
if HasIsIntegersForHomalg( r ) and IsIntegersForHomalg( r ) then
return ( ( Denominator( p ) / HomalgRing( p ) ) * p ) / R;
fi;
fi;
p!.Numerator := Numerator( EvalRingElement( p ) ) / R;
fi;
return p!.Numerator;
end );
##
InstallMethod( Denominator,
"for homalg fake local matrices",
[ IsElementOfHomalgFakeLocalRingRep ],
function( p )
local R, r, q, coeffs;
R := AssociatedGlobalRing( p );
if IsZero( p ) then
return One( R );
fi;
if not IsBound( p!.Denominator ) then
if HasCoefficientsRing( R ) then
r := CoefficientsRing( R );
if HasIsIntegersForHomalg( r ) and IsIntegersForHomalg( r ) then
coeffs := EntriesOfHomalgMatrix( Coefficients( EvalRingElement( p ) ) );
q := CoefficientsRing( AssociatedComputationRing( p ) );
if HasBaseRing( R ) then
r := BaseRing( R );
fi;
coeffs := List( coeffs, c -> Denominator( c / q ) / r );
return Lcm_UsingCayleyDeterminant( coeffs ) / R;
fi;
fi;
p!.Denominator := Denominator( EvalRingElement( p ) ) / R;
fi;
return p!.Denominator;
end );
##
InstallMethod( Numerator,
"for homalg fake local matrices",
[ IsMatrixOverHomalgFakeLocalRingRep ],
function( M )
local R, globalR, fracR, L, DenomList, lcmDenom;
R := HomalgRing( M );
globalR := AssociatedGlobalRing( R );
fracR := AssociatedComputationRing( R );
if not IsBound( M!.Numerator ) then
L := EntriesOfHomalgMatrix( M );
DenomList := List( L, Denominator );
DenomList := Filtered( DenomList, a -> not IsZero( a ) );
lcmDenom := Lcm_UsingCayleyDeterminant( DenomList );
M!.Numerator := globalR * ( ( lcmDenom / fracR ) * Eval( M ) );
M!.Denominator := lcmDenom;
fi;
return M!.Numerator;
end );
##
InstallMethod( Name,
"for homalg fake local ring elements",
[ IsElementOfHomalgFakeLocalRingRep ],
function( o )
local name;
if IsHomalgInternalRingRep( AssociatedComputationRing( o ) ) then
name := String;
else
name := Name;
fi;
return name( EvalRingElement( o ) );
end );
##
InstallMethod( String,
"for homalg fake local ring elements",
[ IsElementOfHomalgFakeLocalRingRep ],
function( o )
return Name( o );
end );
##
InstallMethod( BlindlyCopyMatrixPropertiesToFakeLocalMatrix, ## under construction
"for homalg matrices",
[ IsHomalgMatrix, IsMatrixOverHomalgFakeLocalRingRep ],
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 );
##
InstallMethod( SetMatElm,
"for homalg local matrices",
[ IsMatrixOverHomalgFakeLocalRingRep and IsMutable, IsPosInt, IsPosInt, IsElementOfHomalgFakeLocalRingRep, IsHomalgFakeLocalRingRep ],
function( M, r, c, s, R )
local N, m, e;
# cR := AssociatedComputationRing( R );
m := Eval( M );
m[ r, c ] := EvalRingElement( s );
M!.Eval := m;
end );
##
InstallMethod( AddToMatElm,
"for homalg local matrices",
[ IsMatrixOverHomalgFakeLocalRingRep and IsMutable, IsPosInt, IsPosInt, IsElementOfHomalgFakeLocalRingRep, IsHomalgFakeLocalRingRep ],
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 ] := EvalRingElement( s );
ResetFilterObj( N, IsInitialIdentityMatrix );
#and add this matrix to M
e := Eval( M + N );
SetIsMutableMatrix( e[1], true );
M!.Eval := e;
end );
##
InstallMethod( MatElmAsString,
"for homalg fake local matrices",
[ IsMatrixOverHomalgFakeLocalRingRep, IsPosInt, IsPosInt, IsHomalgFakeLocalRingRep ],
function( M, r, c, R )
local m;
m := Eval( M );
return MatElmAsString( m, r, c, AssociatedComputationRing( R ) );
end );
##
InstallMethod( MatElm,
"for matrices over homalg fake local ring",
[ IsMatrixOverHomalgFakeLocalRingRep, IsPosInt, IsPosInt, IsHomalgFakeLocalRingRep ],
function( M, r, c, R )
local m;
m := Eval( M );
return ElementOfHomalgFakeLocalRing( MatElm( m, r, c, AssociatedComputationRing( R ) ), R );
end );
##
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 fake local rings",
[ IsString, IsHomalgMatrix, IsHomalgFakeLocalRingRep ],
function( filename, M, R )
local ComputationRing, MatStr;
if LoadPackage( "HomalgToCAS" ) <> true then
Error( "the package HomalgToCAS failed to load\n" );
fi;
MatStr := Concatenation( filename, "_matrix" );
ComputationRing := AssociatedComputationRing( M );
SaveHomalgMatrixToFile( MatStr, Eval( M ), ComputationRing );
return MatStr;
end );
##
InstallMethod( LoadHomalgMatrixFromFile,
"for fake local rings",
[ IsString, IsInt, IsInt, IsHomalgFakeLocalRingRep ],
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 MatrixOverHomalgFakeLocalRing( numer, denom, R );
end );
##
InstallMethod( CreateHomalgMatrixFromString,
"constructor for homalg matrices over fake local rings",
[ IsString, IsInt, IsInt, IsHomalgFakeLocalRingRep ],
function( s, r, c, R )
local mat;
mat := CreateHomalgMatrixFromString( s, r, c, AssociatedComputationRing( R ) );
return R * mat;
end );
##
InstallMethod( SetRingProperties,
"for homalg fake local rings",
[ IsHomalgRing ],
function( S )
local R, fracR;
R := AssociatedGlobalRing( S );
fracR := AssociatedComputationRing( S );
if HasIsCommutative( R ) and IsCommutative( R ) then
SetIsCommutative( S, true );
fi;
if HasCoefficientsRing( R ) then
SetCoefficientsRing( S, CoefficientsRing( R ) );
fi;
if HasIsFreePolynomialRing( R ) and IsFreePolynomialRing( R ) then
SetIsIntegralDomain( S, true );
SetIsFreePolynomialRing( S, true );
fi;
if HasIndeterminatesOfPolynomialRing( fracR ) then
SetIndeterminatesOfPolynomialRing( S, List( Indeterminates( fracR ), a -> a / S ) );
fi;
end );
####################################
#
# constructor functions and methods:
#
####################################
##
InstallMethod( LocalizeBaseRingAtPrime,
"constructor for homalg localized rings",
# [ IsHomalgRing and IsCommutative, IsList, IsFinitelyPresentedSubmoduleRep and ConstructedAsAnIdeal and IsPrimeIdeal ],
[ IsHomalgRing and IsCommutative, IsList, IsList ],
function( globalR, X, p )
local indets, Y, k, Q, fracR, RP, localR, baseR, n_gens, gens;
indets := Indeterminates( globalR );
if not IsSubset( indets, X ) then
Error( "the second argument should be a subset of the list of indeterminates of the ring\n" );
fi;
Y := Filtered( indets, a -> not a in X );
k := CoefficientsRing( globalR );
if IsBound( globalR!.PartialQuotientRing ) then
fracR := globalR!.PartialQuotientRing;
else
if HasIsIntegersForHomalg( k ) and IsIntegersForHomalg( k ) then
Q := FieldOfFractions( k );
else
Q := k;
fi;
if not X = [ ] then
Q := AddRationalParameters( Q, X );
fi;
fracR := Q * Y;
fi;
RP := CreateHomalgTableForLocalizedRingsAtPrimeIdeals( fracR );
if not LoadPackage( "RingsForHomalg" ) = true then
Error( "the package RingsForHomalg failed to load\n" );
fi;
if ValueGlobal( "IsHomalgExternalRingInSingularRep" )( globalR ) then
UpdateMacrosOfLaunchedCAS( FakeLocalizeRingMacrosForSingular, homalgStream( globalR ) );
AppendToAhomalgTable( RP, CommonHomalgTableForHomalgFakeLocalRing );
if HasBaseRing( globalR ) then
baseR := BaseRing( globalR );
else
baseR := CoefficientsRing( globalR );
fi;
if HasIsIntegersForHomalg( k ) and IsIntegersForHomalg( k ) then
Unbind( homalgTable( fracR )!.CopyMatrix );
Unbind( homalgTable( fracR )!.CopyElement );
Unbind( homalgTable( globalR )!.CopyMatrix );
Unbind( homalgTable( globalR )!.CopyElement );
Unbind( homalgTable( baseR )!.CopyMatrix );
Unbind( homalgTable( baseR )!.CopyElement );
fi;
fi;
## create the local ring
localR := CreateHomalgRing( globalR, [ TheTypeHomalgFakeLocalRing, TheTypeMatrixOverHomalgFakeLocalRing ], ElementOfHomalgFakeLocalRing, RP );
if not IsString( p ) then
p := List( p,
function( x )
if not IsString( x ) then
return String( x );
fi;
return x;
end );
p := JoinStringsWithSeparator( p );
fi;
p := ParseListOfIndeterminates( SplitString( p, "," ) );
if ForAny( p, IsString ) then
p :=
List( p,
function( x )
if IsString( x ) then
return x / globalR;
fi;
return x;
end );
fi;
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 IsElementOfHomalgFakeLocalRingRep( arg[1][1] ) then
r := arg[1][1];
# return MatrixOverHomalgFakeLocalRing( HomalgMatrix( [ Numerator( r ) ], 1, 1, AssociatedComputationRing( R ) ), Denominator( r ), R );
return MatrixOverHomalgFakeLocalRing( HomalgMatrix( [ r ], 1, 1, AssociatedComputationRing( R ) ), R );
fi;
ar := List( arg,
function( i )
if IsHomalgFakeLocalRingRep( i ) then
return AssociatedComputationRing( i );
else
return i;
fi;
end );
M := CallFuncList( HomalgMatrix, ar );
return MatrixOverHomalgFakeLocalRing( M, R );
end );
## for the display/view methods:
## <A homalg fake local ring>
## <A matrix over a fake local ring>
localR!.description := " (fake) local";
baseR := CoefficientsRing( globalR ) * X;
SetBaseRing(localR, baseR);
#Set the ideal, at which we localize
n_gens := Length( p );
gens := HomalgMatrix( p, n_gens, 1, globalR );
SetGeneratorsOfPrimeIdeal( localR, gens );
localR!.AssociatedGlobalRing := globalR;
localR!.AssociatedComputationRing := fracR;
localR!.AssociatedResidueClassRing := globalR / LeftSubmodule( p );
SetRingProperties( localR );
Perform( X, function( u ) u!.IsUnit := true; end );
return localR;
end );
##
InstallMethod( LocalizeBaseRingAtPrime,
"constructor for homalg localized rings",
# [ IsHomalgRing and IsCommutative, IsList, IsFinitelyPresentedSubmoduleRep and ConstructedAsAnIdeal and IsPrimeIdeal ],
[ IsHomalgRing and IsCommutative, IsList ],
function( R, p )
local indets, X, Rp;
if HasBaseRing( R ) then
indets := RelativeIndeterminatesOfPolynomialRing( R );
X := Filtered( Indeterminates( R ), a -> not a in indets );
else
X := [ ];
fi;
Rp := LocalizeBaseRingAtPrime( R, X, p );
if not IsBound( R!.PartialQuotientRing ) then
R!.PartialQuotientRing := AssociatedComputationRing( Rp );
fi;
return Rp;
end );
##
InstallGlobalFunction( ElementOfHomalgFakeLocalRing,
function( arg )
local nargs, elm, ring, ar, properties, denom, computationring, r;
nargs := Length( arg );
if nargs = 0 then
Error( "empty input\n" );
fi;
elm := arg[1];
if IsElementOfHomalgFakeLocalRingRep( elm ) then
##a local ring element as first argument will just be returned
return elm;
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 );
else
Error( "this argument (now assigned to ar) should be in { IsHomalgRing, IsList( IsFilter ), IsString }\n" );
fi;
od;
computationring := AssociatedComputationRing( ring );
if not IsHomalgRingElement( elm ) then
elm := HomalgRingElement( elm, computationring );
fi;
if IsBound( ring ) then
r := rec( ring := ring );
## Objectify:
ObjectifyWithAttributes( r, TheTypeElementOfHomalgFakeLocalRing, EvalRingElement, elm );
# ObjectifyWithAttributes( r, TheTypeElementOfHomalgFakeLocalRing, ring, ring );
# Objectify( TheTypeElementOfHomalgFakeLocalRing, elm );
fi;
if properties <> [ ] then
for ar in properties do
Setter( ar )( r, true );
od;
fi;
return r;
end );
##
InstallMethod( PolynomialRing,
"for homalg ring elements",
[ IsHomalgFakeLocalRingRep, IsList ],
function( R, indets )
local globalR, baseR, base_indets, p, list, newGlobalR, XX, newLocalR;
#if indets = "" then
# return R;
#fi;
globalR := AssociatedGlobalRing( R );
if HasBaseRing( globalR ) then
baseR := BaseRing ( globalR );
base_indets := List( Indeterminates( baseR ), String );
else
baseR := CoefficientsRing( globalR );
base_indets := [ ];
fi;
p := EntriesOfHomalgMatrix( GeneratorsOfPrimeIdeal( R ) );
list := Concatenation( Filtered( List( Indeterminates( globalR ), String ), a -> not a in base_indets ), indets );
newGlobalR := PolynomialRing( baseR, list );
XX := List( base_indets, x -> x / newGlobalR );
newLocalR := LocalizeBaseRingAtPrime( newGlobalR, XX, p );
SetBaseRing( newLocalR, R );
return newLocalR;
end);
##
InstallMethod( PolynomialRing,
"for homalg ring elements",
[ IsHomalgLocalRingRep, IsList ],
function( R, indets )
local globalR, baseR, p, list, newGlobalR, XX, newLocalR;
globalR := AssociatedGlobalRing( R );
if HasBaseRing( globalR ) then
baseR := BaseRing ( globalR );
else
baseR := globalR;
fi;
p := EntriesOfHomalgMatrix( GeneratorsOfMaximalLeftIdeal( R ) );
list := Concatenation( Filtered( List( Indeterminates( globalR ), x -> String( x ) ), a -> not a in List( Indeterminates( baseR ), x -> String( x ) ) ), indets );
newGlobalR := PolynomialRing( baseR, list );
XX := List( Indeterminates( baseR ), x -> x / newGlobalR );
newLocalR := LocalizeBaseRingAtPrime( newGlobalR, XX, p );
SetBaseRing( newLocalR, R );
return newLocalR;
end);
InstallMethod( IsRightInvertibleMatrix,
"for homalg ring elements",
[ IsMatrixOverHomalgFakeLocalRingRep ],
function( M )
local R;
if IsRightInvertibleMatrix( Numerator( M ) ) then
return true;
fi;
TryNextMethod( );
end );
##
InstallMethod( \/,
"for homalg ring elements",
[ IsRingElement, IsElementOfHomalgFakeLocalRingRep ],
function( a, u )
local R, RP, au;
R := HomalgRing( u );
if not IsHomalgRingElement( a ) then
a := a / R;
fi;
if not HasRingElementConstructor( R ) then
Error( "no ring element constructor found in the ring\n" );
fi;
RP := homalgTable( R );
if IsBound(RP!.DivideByUnit) then
if not IsUnit( u ) then
return fail;
fi;
au := RP!.DivideByUnit( a, u );
if au = fail then
return fail;
fi;
return RingElementConstructor( R )( au, R );
fi;
TryNextMethod( );
end );
##
InstallMethod( \/,
"for homalg ring elements",
[ IsElementOfHomalgFakeLocalRingRep, IsElementOfHomalgFakeLocalRingRep ],
function( a, u )
local R, RP, au;
R := HomalgRing( u );
if not HasRingElementConstructor( R ) then
Error( "no ring element constructor found in the ring\n" );
fi;
RP := homalgTable( R );
if IsBound(RP!.DivideByUnit) then
if not IsUnit( u ) then
return fail;
fi;
au := RP!.DivideByUnit( a, u );
if au = fail then
return fail;
fi;
return RingElementConstructor( R )( au, R );
fi;
TryNextMethod( );
end );
##
InstallMethod( MatrixOverHomalgFakeLocalRing,
"constructor for matrices over fake local rings",
[ IsHomalgMatrix, IsHomalgFakeLocalRingRep ],
function( A, R )
local G, type, matrix, HookDenom, ComputationRing, rr, AA;
G := HomalgRing( A );
ComputationRing := AssociatedComputationRing( R );
if not IsIdenticalObj( ComputationRing , HomalgRing( A ) ) then
AA := ComputationRing * A;
else
AA := A;
fi;
matrix := rec( ring := R );
ObjectifyWithAttributes(
matrix, TheTypeMatrixOverHomalgFakeLocalRing,
Eval, AA
);
BlindlyCopyMatrixPropertiesToFakeLocalMatrix( A, matrix );
return matrix;
end );
##
InstallMethod( \*,
"for homalg matrices",
[ IsHomalgFakeLocalRingRep, IsHomalgMatrix ],
function( R, m )
if IsMatrixOverHomalgFakeLocalRingRep( m ) then
TryNextMethod( );
fi;
return MatrixOverHomalgFakeLocalRing( AssociatedComputationRing( R ) * m, R );
end );
##
InstallMethod( \*,
"for matrices over fake local rings",
[ IsHomalgRing, IsMatrixOverHomalgFakeLocalRingRep ],
function( R, m )
return R * Eval( m );
end );
##
InstallMethod( PostMakeImmutable,
"for matrices over homalg fake local rings ",
[ IsMatrixOverHomalgFakeLocalRingRep and HasEval ],
function( A )
MakeImmutable( Eval( A ) );
end );
##
InstallMethod( SetIsMutableMatrix,
"for matrices over homalg fake local rings ",
[ IsMatrixOverHomalgFakeLocalRingRep, IsBool ],
function( A, b )
if b = true then
SetFilterObj( A, IsMutable );
else
ResetFilterObj( A, IsMutable );
fi;
SetIsMutableMatrix( Eval( A ), b );
end );
##
InstallMethod( RingOfDerivations,
"for fake local rings",
[ IsHomalgFakeLocalRingRep ],
function( Rm )
local R;
R := AssociatedGlobalRing( Rm );
return RingOfDerivations( R );
end );
##
InstallMethod( Coefficients,
"for homalg ring elements",
[ IsElementOfHomalgFakeLocalRingRep ],
function( pol )
local coeffs, monom, S;
coeffs := Coefficients( EvalRingElement( pol ) );
monom := coeffs!.monomials;
S := HomalgRing( pol );
coeffs := S * coeffs;
coeffs!.monomials := List( monom, a -> a / S );
return coeffs;
end );
##
InstallMethod( Random,
"for homalg ring elements",
[ IsHomalgFakeLocalRingRep, IsList ],
function( R, L )
local globalR, baseR, f, g;
globalR := AssociatedGlobalRing( R );
if not HasBaseRing( globalR ) then
Error( );
fi;
baseR := BaseRing( AssociatedGlobalRing( R ) );
f := Random( globalR, L );
f := f / R;
repeat
g := Random( baseR ) + Random( baseR, 0 );
g := g / R;
until IsUnit( g );
return f / g;
end );
##
InstallMethod( Value,
"for a homalg matrix over a fake local ring and two lists",
[ IsMatrixOverHomalgFakeLocalRingRep, IsList, IsList ],
function( M, V, O )
local R, fracR;
R := HomalgRing( M );
fracR := AssociatedComputationRing( R );
V := List( V, i -> i / fracR );
O := List( O, i -> i / fracR );
return R * Value( Eval( M ), V, O );
end );
####################################
#
# View, Print, and Display methods:
#
####################################
##
InstallMethod( Display,
"for homalg fake local ring elements",
[ IsElementOfHomalgFakeLocalRingRep ],
function( r )
Print( Name( r ), "\n" );
end );
##
InstallMethod( Display,
"for homalg matrices over a homalg fake local ring",
[ IsMatrixOverHomalgFakeLocalRingRep ],
function( A )
Display( Eval( A ) );
end );
[ Dauer der Verarbeitung: 0.13 Sekunden
(vorverarbeitet)
]
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