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# SPDX-License-Identifier: GPL-2.0-or-later
# GradedModules: A homalg based package for the Abelian category of finitely presented graded modules over computable graded rings
#
# Implementations
#
## Implementations of procedures for the pair of adjoint Tate functors.
##
## Removes the morphism of lowest degree in the complex T and instead adds another (minimal) morphism there having the same image.
InstallMethod( MinimizeLowestDegreeMorphism,
"for homalg modules",
[ IsCocomplexOfFinitelyPresentedObjectsRep ],
function( T )
local phi, N, TT;
phi:= ImageObjectEmb( LowestDegreeMorphism( T ) );
N := Source( phi );
ByASmallerPresentation( N );
phi := PreCompose( CokernelEpi( PresentationMorphism( N ) ), phi );
if Length( ObjectDegreesOfComplex( T ) ) = 2 then
TT := HomalgCocomplex( phi, LowestDegree( T ) );
else
TT := Subcomplex( T, LowestDegree( T ) + 1, HighestDegree( T ) );
Add( phi, TT );
fi;
return TT;
end );
####################################
#
# methods for operations:
#
####################################
## <#GAPDoc Label="TateResolution">
## <ManSection>
## <Oper Arg="M, degree_lowest, degree_highest" Name="TateResolution"/>
## <Returns>a &homalg; cocomplex</Returns>
## <Description>
## Compute the Tate resolution of the sheaf <A>M</A>.
## <#Include Label="TateResolution:example1">
## In the following we construct the different exterior powers of the cotangent bundle
## shifted by <M>1</M>. Observe how a single <M>1</M> travels along the diagnoal
## in the window <M>[ -3 .. 0 ] x [ 0 .. 3 ]</M>. <Br/><Br/>
## First we start with the structure sheaf with its Tate resolution:
## <#Include Label="TateResolution:example2">
## The Castelnuovo-Mumford regularity of the <E>underlying module</E> is distinguished
## among the list of twists by the character <C>'V'</C> pointing to it. It is <E>not</E>
## an invariant of the sheaf (see the next diagram). <Br/><Br/>
## The residue class field (i.e. S modulo the maximal homogeneous ideal):
## <#Include Label="TateResolution:example3">
## Another way of constructing the structure sheaf:
## <#Include Label="TateResolution:example4">
## The cotangent bundle:
## <#Include Label="TateResolution:example5">
## the cotangent bundle shifted by <M>1</M> with its Tate resolution:
## <#Include Label="TateResolution:example6">
## The second power <M>U^2</M> of the shifted cotangent bundle <M>U=U^1</M> and its Tate resolution:
## <#Include Label="TateResolution:example7">
## The third power <M>U^3</M> of the shifted cotangent bundle <M>U=U^1</M> and its Tate resolution:
## <#Include Label="TateResolution:example8">
## Another way to construct <M>U^2=U^(3-1)</M>:
## <#Include Label="TateResolution:example9">
## </Description>
## </ManSection>
## <#/GAPDoc>
## gap> Display(S);
## Q[x0,x1,x2,x3]
## gap> Display(A);
## Q{e0,e1,e2,e3}
## gap> Display(cotangent);
## x1,-x2,x3,0, 0, 0,
## x0,0, 0, -x2,x3, 0,
## 0, x0, 0, -x1,0, x3,
## 0, 0, x0,0, -x1,x2
##
## (graded, generators degrees: [ 2, 2, 2, 2, 2, 2 ])
##
## Cokernel of the map
##
## R^(1x4) --> R^(1x6), ( for R := Q[x0,x1,x2,x3] )
##
## currently represented by the above matrix
##
InstallGlobalFunction( _Functor_TateResolution_OnGradedModules, ### defines: TateResolution (object part)
function( l, _M )
local A, degree_lowest, degree_highest, M, CM, p, d_low, d_high, T, positions,
old_degrees, old_T, tate, i, K, Kres, degrees, CM_sheaf, result;
if not Length( l ) = 3 then
Error( "wrong number of elements in zeroth parameter, expected an exterior algebra and two integers" );
else
A := l[1];
degree_lowest := l[2];
degree_highest := l[3];
if not ( IsHomalgRing( A ) and IsExteriorRing( A ) and IsInt( degree_lowest ) and IsInt( degree_highest ) ) then
Error( "wrong number of elements in zeroth parameter, expected an exterior algebra and two integers" );
fi;
fi;
if IsHomalgRing( _M ) then
M := FreeRightModuleWithDegrees( 1, _M );
else
M := _M;
fi;
CM := ValueOption( "CM" );
if not IsInt( CM ) then
CM := CastelnuovoMumfordRegularity( M );
CM := HomalgElementToInteger( CM );
fi;
if not IsBound( M!.TateResolution ) then
M!.TateResolution := rec( );
fi;
p := PositionOfTheDefaultPresentation( M );
if IsGradedModuleRep( M ) and IsBound( M!.TateResolution!.(p) ) then
T := M!.TateResolution!.(p);
old_T := ShallowCopy( ObjectsOfComplex( T ) );
positions := List( old_T, PositionOfTheDefaultPresentation );
old_degrees := ObjectDegreesOfComplex( T );
old_degrees := List( old_degrees, HomalgElementToInteger );
tate := HighestDegreeMorphism( T );
d_high := old_degrees[ Length( old_degrees ) ] - 1;
d_low := old_degrees[ 1 ];
# restore the original presentations
# (Perform needs a return value)
Perform( old_T, function( a ) SetPositionOfTheDefaultPresentation( a, 1 ); return true; end );
else
d_high := Maximum( CM, degree_lowest );
d_low := d_high;
tate := RepresentationMapOfKoszulId( d_high, M, A );
T := HomalgCocomplex( tate, d_high );
old_T := ShallowCopy( ObjectsOfComplex( T ) );
positions := List( ObjectsOfComplex( T ), PositionOfTheDefaultPresentation );
fi;
## above the Castelnuovo-Mumford regularity
for i in [ d_high + 1 .. degree_highest - 1 ] do
## the Koszul map has linear entries by construction
tate := RepresentationMapOfKoszulId( i, M, A );
Add( T, tate );
od;
## below the Castelnuovo-Mumford regularity
if degree_lowest < d_low then
# The morphism of lowest degree is not part of a minimal complex.
# However, its image is the kernel of the following map.
# Thus, we find a minimal map having the same image.
T := MinimizeLowestDegreeMorphism( T );
tate := LowestDegreeMorphism( T );
K := Kernel( tate );
## get rid of the units in the presentation of the kernel K
ByASmallerPresentation( K );
Kres := Resolution( d_low - degree_lowest, K );
tate := PreCompose( CoveringEpi( K ), KernelEmb( tate ) );
Add( tate, T );
for i in [ 1 .. d_low - degree_lowest - 1 ] do
tate := CertainMorphism( Kres, i );
Add( tate, T );
od;
if not HasCastelnuovoMumfordRegularityOfSheafification( M ) then
old_degrees := Set( DegreesOfGenerators( CertainObject( T, d_low ) ) );
Assert( 0, Length( old_degrees ) <= 1 );
if Length( old_degrees ) = 1 then
for i in Reversed( [ degree_lowest .. d_low - 1 ] ) do
degrees := DegreesOfGenerators( CertainObject( T, i ) );
if Minimum( degrees ) < Minimum( old_degrees ) - 1 then
SetCastelnuovoMumfordRegularityOfSheafification( M, i + 1 );
break;
fi;
old_degrees := degrees;
od;
fi;
fi;
fi;
## check assertion
Assert( 3, IsAcyclic( T ) );
SetIsAcyclic( T, true );
## pass some options to the operation BettiTable (applied on complexes):
T!.display_twist := true;
T!.EulerCharacteristic := HilbertPolynomial( M );
## starting from the Castelnuovo-Mumford regularity
## (and going right) all higher cohomologies vanish
if HasCastelnuovoMumfordRegularityOfSheafification( M ) then
CM_sheaf := CastelnuovoMumfordRegularityOfSheafification( M );
T!.higher_vanish := CM_sheaf;
T!.markers := [ [ CM_sheaf, "S" ], [ CM, "M" ] ];
else
T!.higher_vanish := CM;
T!.markers := [ [ CM, "M" ] ];
fi;
if IsGradedModuleRep( M ) then
M!.TateResolution!.(p) := T;
fi;
result := Subcomplex( T, degree_lowest, degree_highest );
## check assertion
Assert( 3, IsAcyclic( result ) );
SetIsAcyclic( result, true );
## pass some options to the operation BettiTable (applied on complexes):
result!.display_twist := true;
result!.EulerCharacteristic := HilbertPolynomial( M );
## starting from the Castelnuovo-Mumford regularity
## (and going right) all higher cohomologies vanish
if HasCastelnuovoMumfordRegularityOfSheafification( M ) then
CM_sheaf := CastelnuovoMumfordRegularityOfSheafification( M );
result!.higher_vanish := CM_sheaf;
result!.markers := [ [ CM_sheaf, "S" ], [ CM, "M" ] ];
else
result!.higher_vanish := CM;
result!.markers := [ [ CM, "M" ] ];
fi;
# restore the presentation from the beginning of the procedure
for i in [ 1 .. Length( old_T ) ] do
SetPositionOfTheDefaultPresentation( old_T[i], positions[i] );
od;
return result;
end );
##
InstallMethod( TateResolution,
"for homalg modules",
[ IsHomalgModule, IsInt, IsInt ],
function( M, degree_lowest, degree_highest )
local A;
A := KoszulDualRing( HomalgRing( M ) );
return TateResolution( M, A, degree_lowest, degree_highest );
end );
##
InstallMethod( TateResolution,
"for homalg elements",
[ IsHomalgModule, IsObject, IsObject ],
function( M, degree_lowest, degree_highest )
return TateResolution( M, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
##
InstallMethod( TateResolution,
"for homalg elements",
[ IsHomalgRing, IsObject, IsObject, IsHomalgRingOrModule ],
function( M, degree_lowest, degree_highest, R )
return TateResolution( M, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ), R );
end );
##
InstallMethod( TateResolution,
"for homalg modules",
[ IsHomalgModule, IsHomalgRing and IsExteriorRing, IsInt, IsInt ],
function( M, A, degree_lowest, degree_highest )
return TateResolution( [ A, degree_lowest, degree_highest ], M );
end );
##
InstallMethod( TateResolution,
"for homalg elements",
[ IsHomalgModule, IsHomalgRing, IsObject, IsObject ],
function( M, R, degree_lowest, degree_highest )
return TateResolution( M, R, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
##
InstallMethod( TateResolution,
"for homalg modules",
[ IsMapOfGradedModulesRep, IsInt, IsInt ],
function( phi, degree_lowest, degree_highest )
local A;
A := KoszulDualRing( HomalgRing( phi ) );
return TateResolution( phi, A, degree_lowest, degree_highest );
end );
##
InstallMethod( TateResolution,
"for homalg elements",
[ IsHomalgGradedMap, IsObject, IsObject ],
function( M, degree_lowest, degree_highest )
return TateResolution( M, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
##
InstallMethod( TateResolution,
"for homalg modules",
[ IsMapOfGradedModulesRep, IsHomalgRing and IsExteriorRing, IsInt, IsInt ],
function( phi, A, degree_lowest, degree_highest )
return TateResolution( [ A, degree_lowest, degree_highest ], phi );
end );
##
InstallMethod( TateResolution,
"for homalg elements",
[ IsHomalgGradedMap, IsHomalgRing, IsObject, IsObject ],
function( M, R, degree_lowest, degree_highest )
return TateResolution( M, R, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
##
InstallGlobalFunction( _Functor_TateResolution_OnGradedMaps, ### defines: TateResolution (morphism part)
function( F_source, F_target, arg_before_pos, phi, arg_behind_pos )
local l, A, degree_lowest, degree_highest, degree_highest2, CM, T_source, T_range, T, T2, ii, i;
l := arg_before_pos[1];
if not Length( l ) = 3 then
Error( "wrong number of elements in zeroth parameter, expected an exterior algebra and two integers" );
else
A := l[1];
degree_lowest := l[2];
degree_highest := l[3];
if not ( IsHomalgRing( A ) and IsExteriorRing( A ) and IsInt( degree_lowest ) and IsInt( degree_highest ) ) then
Error( "wrong number of elements in 0th parameter, expected an exterior algebra and two integers" );
fi;
fi;
CM := ValueOption( "CM" );
if not IsInt( CM ) then
CM := CastelnuovoMumfordRegularity( phi );
CM := HomalgElementToInteger( CM );
fi;
degree_highest2 := Maximum( degree_highest, CM + 1 );
# we need to compute the module down from the CastelnuovoMumfordRegularity
T_source := TateResolution( Source( phi ), A, degree_lowest, degree_highest2 );
T_range := TateResolution( Range( phi ), A, degree_lowest, degree_highest2 );
i := degree_highest2;
T2 := HomalgChainMorphism( A * HomogeneousPartOverCoefficientsRing( i, phi ), T_source, T_range, i );
if degree_highest2 = degree_highest then
T := HomalgChainMorphism( LowestDegreeMorphism( T2 ), F_source, F_target, i );
fi;
for ii in [ degree_lowest .. degree_highest2 - 1 ] do
i := ( degree_highest2 - 1 ) + degree_lowest - ii;
Add( CompleteImageSquare( CertainMorphism( T_source, i ), LowestDegreeMorphism( T2 ), CertainMorphism( T_range, i ) ), T2 );
if i <= degree_highest and not IsBound( T ) then
T := HomalgChainMorphism( LowestDegreeMorphism( T2 ), F_source, F_target, i );
elif i < degree_highest then
Add( LowestDegreeMorphism( T2 ), T );
fi;
od;
return T;
end );
BindGlobal( "Functor_TateResolution_ForGradedModules",
CreateHomalgFunctor(
[ "name", "TateResolution" ],
[ "category", HOMALG_GRADED_MODULES.category ],
[ "operation", "TateResolution" ],
[ "number_of_arguments", 1 ],
[ "0", [ IsList ] ],
[ "1", [ [ "covariant", "left adjoint", "distinguished" ], HOMALG_GRADED_MODULES.FunctorOn ] ],
[ "OnObjects", _Functor_TateResolution_OnGradedModules ],
[ "OnMorphisms", _Functor_TateResolution_OnGradedMaps ]
)
);
Functor_TateResolution_ForGradedModules!.ContainerForWeakPointersOnComputedBasicObjects := true;
Functor_TateResolution_ForGradedModules!.ContainerForWeakPointersOnComputedBasicMorphisms := true;
InstallFunctor( Functor_TateResolution_ForGradedModules );
## written for Sepp's talk, it has to become functorial in the future
InstallMethod( TateResolution,
"for homalg modules",
[ IsHomalgModule, IsInt, IsInt ],
function( L, degree_lowest, degree_highest )
local A, mu0, tate, inj, mu;
A := HomalgRing( L );
if not ( HasIsExteriorRing( A ) and IsExteriorRing( A ) ) then
TryNextMethod( );
fi;
mu0 := FirstMorphismOfResolution( L );
tate := HomalgCocomplex( mu0 );
CompleteComplexByResolution( -degree_lowest, tate );
if degree_highest < 2 then
return tate;
fi;
inj := HomalgComplex( GradedHom( mu0 ) );
inj := GradedHom( CompleteComplexByResolution( degree_highest - 1, inj ) );
inj := MorphismsOfComplex( inj );
inj := inj{[ 2 .. Length( inj ) ]};
inj[1] := PreCompose( NatTrIdToHomHom_R( Range( mu0 ) ), inj[1] );
for mu in inj do
Add( tate, mu );
od;
return tate;
end );
##
## LinearStrandOfTateResolution
##
InstallMethod( ResolveLinearly,
"for homalg cocomplexes",
[ IsInt, IsHomalgComplex, IsInt ],
function( n, T, shift )
local know_regularity, i, tate, K, deg, certain_deg, phi, regularity;
know_regularity := false;
for i in [ 1 .. n ] do
tate := LowestDegreeMorphism( T );
K := Kernel( tate );
## get rid of the units in the presentation of the kernel K
ByASmallerPresentation( K );
tate := PreCompose( CoveringEpi( K ), KernelEmb( tate ) );
# phi is the embedding of the right degree into the module
deg := DegreesOfGenerators( Source( tate ) );
deg := List( deg, HomalgElementToInteger );
certain_deg := Filtered( [ 1 .. Length( deg ) ], a -> deg[a] = Minimum( ObjectDegreesOfComplex( T ) ) - 1 + shift );
if [ 1 .. Length( deg ) ] <> certain_deg then
if not know_regularity then
regularity := Minimum( ObjectDegreesOfComplex( T ) );
know_regularity := true;
fi;
if IsHomalgLeftObjectOrMorphismOfLeftObjects( tate ) then
phi := GradedMap( CertainRows( HomalgIdentityMatrix( NrGenerators( Source( tate ) ), HomalgRing( tate ) ), certain_deg ), "free", Source( tate ) );
else
phi := GradedMap( CertainColumns( HomalgIdentityMatrix( NrGenerators( Source( tate ) ), HomalgRing( tate ) ), certain_deg ), "free", Source( tate ) );
fi;
Assert( 4, IsMorphism( phi ) );
SetIsMorphism( phi, true );
tate := PreCompose( phi, tate );
fi;
Assert( 3, IsMorphism( tate ) );
SetIsMorphism( tate, true );
Add( tate, T );
od;
if know_regularity then
return regularity;
else
return fail;
fi;
end );
InstallMethod( ResolveLinearly,
"for homalg cocomplexes",
[ IsInt, IsHomalgComplex ],
function( n, T )
return ResolveLinearly( n, T, 0 );
end );
##
InstallGlobalFunction( _Functor_LinearStrandOfTateResolution_OnGradedModules, ### defines: StrandOfTateResolution (object part)
function( l, _M )
local A, degree_lowest, degree_highest, n, M, CM, p, d_low, d_high, tate, T, i, know_regularity, ii, K, deg, certain_deg, phi, regularity, result;
if not Length( l ) = 3 then
Error( "wrong number of elements in zeroth parameter, expected an exterior algebra and two integers" );
else
A := l[1];
degree_lowest := l[2];
degree_highest := l[3];
if not ( IsHomalgRing( A ) and IsExteriorRing( A ) and IsInt( degree_lowest ) and IsInt( degree_highest ) )then
Error( "wrong number of elements in zeroth parameter, expected an exterior algebra and two integers." );
fi;
fi;
n := Length( IndeterminateAntiCommutingVariablesOfExteriorRing( A ) );
if IsHomalgRing( _M ) then
M := FreeRightModuleWithDegrees( 1, _M );
else
M := _M;
fi;
CM := ValueOption( "CM" );
if not IsInt( CM ) then
CM := CastelnuovoMumfordRegularity( M );
CM := HomalgElementToInteger( CM );
fi;
p := PositionOfTheDefaultPresentation( M );
if not IsBound( M!.LinearStrandOfTateResolution ) then
M!.LinearStrandOfTateResolution := rec( );
fi;
if IsGradedModuleRep( M ) and IsBound( M!.LinearStrandOfTateResolution!.(p) ) then
T := M!.LinearStrandOfTateResolution!.(p)[2];
tate := HighestDegreeMorphism( T );
d_high := T!.degrees[ Length( T!.degrees ) ] - 1;
d_low := T!.degrees[ 1 ];
regularity := M!.LinearStrandOfTateResolution!.(p)[1];
if IsInt( regularity ) then
know_regularity := true;
else
know_regularity := false;
fi;
else
d_high := Maximum( CM, degree_lowest );
d_low := d_high;
tate := RepresentationMapOfKoszulId( d_high, M, A );
T := HomalgCocomplex( tate, d_high );
know_regularity := false;
fi;
## above the Castelnuovo-Mumford regularity
for i in [ d_high + 1 .. degree_highest - 1 ] do
## the Koszul map has linear entries by construction
tate := RepresentationMapOfKoszulId( i, M, A );
Add( T, tate );
od;
## below the Castelnuovo-Mumford regularity
if degree_lowest < d_low then
# The morphism of lowest degree is not part of a minimal complex.
# However, its image is the kernel of the following map.
# Thus, we find a minimal map having the same image.
if LowestDegree( T ) = CM then
T := MinimizeLowestDegreeMorphism( T );
fi;
if know_regularity then
ResolveLinearly( d_low - degree_lowest, T, n );
else
regularity := ResolveLinearly( d_low - degree_lowest, T, n );
fi;
if regularity = fail then
know_regularity := false;
else
know_regularity := true;
fi;
fi;
## pass some options to the operation BettiTable (applied on complexes):
T!.display_twist := true;
## starting from the Castelnuovo-Mumford regularity
## (and going right) all higher cohomologies vanish
T!.higher_vanish := CM;
if IsGradedModuleRep( M ) then
M!.LinearStrandOfTateResolution!.(p) := [ fail, T ];
fi;
result := Subcomplex( T, degree_lowest, degree_highest );
## pass some options to the operation BettiTable (applied on complexes):
result!.display_twist := true;
## starting from the Castelnuovo-Mumford regularity
## (and going right) all higher cohomologies vanish
result!.higher_vanish := CM;
if know_regularity then
result!.regularity := Maximum( HOMALG_GRADED_MODULES!.LowerTruncationBound, regularity );
M!.LinearStrandOfTateResolution!.(p)[1] := result!.regularity;
else
result!.regularity := degree_lowest;
fi;
Assert( 3, IsComplex( result ) );
SetIsComplex( result, true );
return result;
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg elements",
[ IsHomalgRing, IsObject, IsObject, IsHomalgRingOrModule ],
function( M, degree_lowest, degree_highest, R )
return LinearStrandOfTateResolution( M, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ), R );
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg modules",
[ IsGradedModuleRep, IsInt, IsInt ],
function( M, degree_lowest, degree_highest )
local A;
A := KoszulDualRing( HomalgRing( M ) );
return LinearStrandOfTateResolution( M, A, degree_lowest, degree_highest );
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg elements",
[ IsGradedModuleRep, IsObject, IsObject ],
function( M, degree_lowest, degree_highest )
return LinearStrandOfTateResolution( M, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg modules",
[ IsGradedModuleRep, IsHomalgRing and IsExteriorRing, IsInt, IsInt ],
function( M, A, degree_lowest, degree_highest )
return LinearStrandOfTateResolution( [ A, degree_lowest, degree_highest ], M );
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg elements",
[ IsHomalgRingOrModule, IsHomalgRing, IsObject, IsObject ],
function( M, R, degree_lowest, degree_highest )
return LinearStrandOfTateResolution( M, R, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg modules",
[ IsMapOfGradedModulesRep, IsInt, IsInt ],
function( phi, degree_lowest, degree_highest )
local A;
A := KoszulDualRing( HomalgRing( phi ) );
return LinearStrandOfTateResolution( phi, A, degree_lowest, degree_highest );
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg modules",
[ IsMapOfGradedModulesRep, IsHomalgRing and IsExteriorRing, IsInt, IsInt ],1,
function( phi, A, degree_lowest, degree_highest )
return LinearStrandOfTateResolution( [ A, degree_lowest, degree_highest ], phi );
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg elements",
[ IsMapOfGradedModulesRep, IsHomalgRing and IsExteriorRing, IsObject, IsObject ],0,
function( M, R, degree_lowest, degree_highest )
return LinearStrandOfTateResolution( M, R, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
##
InstallGlobalFunction( _Functor_LinearStrandOfTateResolution_OnGradedMaps, ### defines: StrandOfTateResolution (morphism part)
function( F_source, F_target, arg_before_pos, phi, arg_behind_pos )
local l, A, degree_lowest, degree_highest, n, degree_highest2, CM, T_source, T_range, T, T2, ii, i;
l := arg_before_pos[1];
if not Length( l ) = 3 then
Error( "wrong number of elements in zeroth parameter, expected an exterior algebra and two integers" );
else
A := l[1];
degree_lowest := l[2];
degree_highest := l[3];
if not ( IsHomalgRing( A ) and IsExteriorRing( A ) and IsInt( degree_lowest ) and IsInt( degree_highest ) ) then
Error( "wrong number of elements in 0th parameter, expected an exterior algebra and two integers" );
fi;
fi;
n := Length( IndeterminateAntiCommutingVariablesOfExteriorRing( A ) );
CM := ValueOption( "CM" );
if not IsInt( CM ) then
CM := CastelnuovoMumfordRegularity( phi );
CM := HomalgElementToInteger( CM );
fi;
degree_highest2 := Maximum( degree_highest, CM + 1 );
# We need to compute the module down from the CastelnuovoMumfordRegularity
# So the Ts are just longer versions of the Fs
T_source := LinearStrandOfTateResolution( Source( phi ), A, degree_lowest, degree_highest2 );
T_range := LinearStrandOfTateResolution( Range( phi ), A, degree_lowest, degree_highest2 );
i := degree_highest2;
T2 := HomogeneousPartOverCoefficientsRing( i, phi );
T2 := A^(-n) * ( A * T2 );
T2 := HomalgChainMorphism( T2, T_source, T_range, i );
if degree_highest2 = degree_highest then
T := HomalgChainMorphism( LowestDegreeMorphism( T2 ), F_source, F_target, i );
fi;
for ii in [ degree_lowest .. degree_highest2 - 1 ] do
i := ( degree_highest2 - 1 ) + degree_lowest - ii;
Add( CompleteImageSquare( CertainMorphism( T_source, i ), LowestDegreeMorphism( T2 ), CertainMorphism( T_range, i ) ), T2 );
if i <= degree_highest and not IsBound( T ) then
T := HomalgChainMorphism( LowestDegreeMorphism( T2 ), F_source, F_target, i );
elif i < degree_highest then
Add( LowestDegreeMorphism( T2 ), T );
fi;
od;
return T;
end );
##
InstallMethod( LinearStrandOfTateResolution,
"for homalg elements",
[ IsHomalgGradedMap, IsObject, IsObject ],
function( M, degree_lowest, degree_highest )
return LinearStrandOfTateResolution( M, HomalgElementToInteger( degree_lowest ), HomalgElementToInteger( degree_highest ) );
end );
BindGlobal( "Functor_LinearStrandOfTateResolution_ForGradedModules",
CreateHomalgFunctor(
[ "name", "LinearStrandOfTateResolution" ],
[ "category", HOMALG_GRADED_MODULES.category ],
[ "operation", "LinearStrandOfTateResolution" ],
[ "number_of_arguments", 1 ],
[ "0", [ IsList ] ],
[ "1", [ [ "covariant", "left adjoint", "distinguished" ], HOMALG_GRADED_MODULES.FunctorOn ] ],
[ "OnObjects", _Functor_LinearStrandOfTateResolution_OnGradedModules ],
[ "OnMorphisms", _Functor_LinearStrandOfTateResolution_OnGradedMaps ]
)
);
Functor_LinearStrandOfTateResolution_ForGradedModules!.ContainerForWeakPointersOnComputedBasicObjects := true;
Functor_LinearStrandOfTateResolution_ForGradedModules!.ContainerForWeakPointersOnComputedBasicMorphisms := true;
InstallFunctor( Functor_LinearStrandOfTateResolution_ForGradedModules );
[ Dauer der Verarbeitung: 0.38 Sekunden
(vorverarbeitet)
]
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