|
# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#
## Implementation stuff for (spectral) filtrations.
##
InstallMethod( FiltrationOfTotalDefect,
"for spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsInt ],
function( E, n )
local gen_embs, bidegrees, degrees, gen_embs_n, pq;
if HasGeneralizedEmbeddingsInTotalDefects( E ) then
gen_embs := GeneralizedEmbeddingsInTotalDefects( E );
bidegrees := BidegreesOfSpectralSequence( E, n );
## in case E is a cohomological spectral sequences
if IsSpectralCosequenceOfFinitelyPresentedObjectsRep( E ) then
bidegrees := Reversed( bidegrees );
fi;
degrees := List( bidegrees, a -> a[1] );
gen_embs_n := rec( degrees := degrees,
bidegrees := bidegrees );
for pq in bidegrees do
if IsBound( gen_embs!.(String( pq )) ) then
gen_embs_n.(String( pq[1] )) := gen_embs!.(String( pq ));
fi;
od;
if IsSpectralSequenceOfFinitelyPresentedObjectsRep( E ) then
return HomalgAscendingFiltration( gen_embs_n, IsFiltration, true );
else
return HomalgDescendingFiltration( gen_embs_n, IsFiltration, true );
fi;
fi;
return fail;
end );
##
InstallMethod( FiltrationOfTotalDefect,
"for spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex ],
function( E )
return FiltrationOfTotalDefect( E, 0 );
end );
##
InstallMethod( FiltrationOfObjectInCollapsedSheetOfTransposedSpectralSequence,
"for spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsInt ],
function( E, n )
local BC, gen_embs, bidegrees, degrees, gen_embs_n, pq;
BC := UnderlyingBicomplex( E );
if IsBound( E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence ) then
if IsFunction( E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence ) then
E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence( );
fi;
gen_embs := E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence;
bidegrees := BidegreesOfSpectralSequence( E, n );
## this is necessary for handling not only homological
## but also cohomological spectral sequences
if IsSpectralCosequenceOfFinitelyPresentedObjectsRep( E ) then
bidegrees := Reversed( bidegrees );
fi;
degrees := List( bidegrees, a -> a[1] );
gen_embs_n := rec( degrees := degrees,
bidegrees := bidegrees );
for pq in bidegrees do
if IsBound( gen_embs!.(String( pq )) ) then
gen_embs_n.(String( pq[1] )) := gen_embs!.(String( pq ));
fi;
od;
if IsSpectralSequenceOfFinitelyPresentedObjectsRep( E ) then
return HomalgAscendingFiltration( gen_embs_n, IsFiltration, true );
else
return HomalgDescendingFiltration( gen_embs_n, IsFiltration, true );
fi;
fi;
return fail;
end );
##
InstallMethod( FiltrationOfObjectInCollapsedSheetOfTransposedSpectralSequence,
"for spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex ],
function( E )
return FiltrationOfObjectInCollapsedSheetOfTransposedSpectralSequence( E, 0 );
end );
##
InstallMethod( FiltrationBySpectralSequence,
"for spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsInt ],
function( E, n )
local BC, filt;
BC := UnderlyingBicomplex( E );
if IsBound( E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence ) then
if IsFunction( E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence ) then
E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence( );
fi;
filt := FiltrationOfObjectInCollapsedSheetOfTransposedSpectralSequence( E, n );
elif HasGeneralizedEmbeddingsInTotalDefects( E ) then
filt := FiltrationOfTotalDefect( E, n );
else
return fail;
fi;
## enrich the filtration with the spectral sequence used to construct it
SetSpectralSequence( filt, E );
return filt;
end );
##
InstallMethod( FiltrationBySpectralSequence,
"for spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex ],
function( E )
return FiltrationBySpectralSequence( E, 0 );
end );
##
InstallMethod( SetAttributesByPurityFiltrationViaBidualizingSpectralSequence,
"for (ascending) purity filtrations",
[ IsFiltrationOfFinitelyPresentedObjectRep and
IsAscendingFiltration and
IsPurityFiltration ],
function( filt )
local II_E, M, non_zero_p, l, p;
if not HasSpectralSequence( filt ) then
Error( "this purity filtration does not have the attribute SpectralSequence;",
"maybe it was not computed using the bidualizing spectral sequence\n" );
fi;
II_E := SpectralSequence( filt );
M := UnderlyingObject( filt );
## set different porperties and attributes of the pure parts
Perform( DegreesOfFiltration( filt ),
function( p )
local L;
L := CertainObject( filt, p );
if not IsZero( L ) then
SetCodegreeOfPurity( L, StaircaseOfStability( II_E, [ p, -p ], 2 ) );
fi;
end );
## set the codegree of purity for the torsion-free factor (in case it is already computed)
if HasTorsionFreeFactorEpi( M ) and CertainObject( II_E, [ 0, 0 ] ) <> fail then
SetCodegreeOfPurity( TorsionFreeFactor( M ), StaircaseOfStability( II_E, [ 0, 0 ], 2 ) );
fi;
non_zero_p := Filtered( DegreesOfFiltration( filt ), p -> not IsZero( CertainObject( filt, p ) ) );
l := Length( non_zero_p );
## if only one graded part is non-trivial, the object M is pure
if l > 0 then
p := non_zero_p[l];
SetGrade( M, -p );
if l = 1 then
SetCodegreeOfPurity( M, StaircaseOfStability( II_E, [ p, -p ], 2 ) );
else
SetCodegreeOfPurity( M, infinity );
fi;
fi;
end );
##
InstallMethod( SetAttributesByPurityFiltration,
"for (ascending) purity filtrations",
[ IsFiltrationOfFinitelyPresentedObjectRep and
IsAscendingFiltration and
IsPurityFiltration ],
function( filt )
local M, degrees, pure_degrees, L, non_zero_p, l, p, M0;
M := UnderlyingObject( filt );
degrees := DegreesOfFiltration( filt );
pure_degrees := degrees{[ 2 .. Length( degrees ) ]};
## set different porperties and attributes of the pure parts
Perform( pure_degrees,
function( p )
local L;
L := CertainObject( filt, p );
if not IsZero( L ) then
SetGrade( L, -p );
SetIsPure( L, true );
fi;
end );
p := degrees[1];
L := CertainObject( filt, p );
if not IsZero( L ) then
SetGrade( L, -p );
if HasIsCompletePurityFiltration( filt ) and
IsCompletePurityFiltration( filt ) then
SetIsPure( L, true );
fi;
fi;
non_zero_p := Filtered( degrees, p -> not IsZero( CertainObject( filt, p ) ) );
l := Length( non_zero_p );
## if only one graded part is non-trivial, the object M is pure
if l > 0 then
p := non_zero_p[l];
SetGrade( M, -p );
if l = 1 then
if HasIsCompletePurityFiltration( filt ) and
IsCompletePurityFiltration( filt ) then
SetIsPure( M, true );
fi;
else
SetIsPure( M, false );
fi;
fi;
M0 := CertainObject( filt, 0 );
## the rank of the 0-th part M0 is the rank of the object M
if HasRankOfObject( M ) and RankOfObject( M ) > 0 then
SetRankOfObject( M0, RankOfObject( M ) );
elif HasRankOfObject( M0 ) then
SetRankOfObject( M, RankOfObject( M0 ) );
fi;
end );
##
InstallMethod( PurityFiltrationViaBidualizingSpectralSequence,
"for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M ) ## does not set the attribute PurityFiltration
local II_E, filt, I_E, iso;
II_E := BidualizingSpectralSequence( M, [ 0 ] );
## the underlying object of this filtraton is L_0( (R^0 F) G )( M )
## and not M
filt := FiltrationBySpectralSequence( II_E );
## construct the isomorphism
## L_0( (R^0 F) G )( M ) -> L_0( FG )( M ) -> FG( M ) -> M:
## the associated first spectral sequence
I_E := AssociatedFirstSpectralSequence( II_E );
## L_0( (R^0 F) G )( M ) -> L_0( FG )( M )
iso := I_E!.NaturalTransformations.(String( [ 0, 0 ] ));
## L_0( (R^0 F) G )( M ) -> L_0( FG )( M ) -> CoveringObject( FG( M ) )
iso := PreCompose( iso, I_E!.NaturalGeneralizedEmbeddings.(String( [ 0, 0 ])) );
## L_0( (R^0 F) G )( M ) -> L_0( FG )( M ) -> CoveringObject( FG( M ) ) -> CoveringObject( M )
iso := iso / NatTrIdToHomHom_R( CoveringObject( M ) ); ## lift
## L_0( (R^0 F) G )( M ) -> L_0( FG )( M ) -> CoveringObject( FG( M ) ) -> CoveringObject( M ) -> M
## finally giving the isomorphism
## L_0( (R^0 F) G )( M ) -> M
iso := PreCompose( iso, CoveringEpi( M ) );
Assert( 3, IsIsomorphism( iso ) );
SetIsIsomorphism( iso, true );
## transfer the known properties/attributes in both directions
UpdateObjectsByMorphism( iso );
## finally compose with the natural isomorphism
## to compute the induced filtraton on M
filt := filt * iso;
## start with this as it might help to find more properties/attributes
ByASmallerPresentation( filt );
SetIsPurityFiltration( filt, true );
SetIsCompletePurityFiltration( filt, true );
SetSpectralSequence( filt, II_E );
SetAttributesByPurityFiltration( filt );
SetAttributesByPurityFiltrationViaBidualizingSpectralSequence( filt );
M!.PurityFiltrationViaBidualizingSpectralSequence := filt;
return filt;
end );
##
InstallMethod( PurityFiltration,
"for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
PurityFiltrationViaBidualizingSpectralSequence );
##
InstallMethod( OnPresentationAdaptedToFiltration,
"for filtrations of homalg static objects",
[ IsFiltrationOfFinitelyPresentedObjectRep ],
function( filt )
local M, pos, iso;
M := UnderlyingObject( filt );
pos := PositionOfTheDefaultPresentation( M );
if IsBound( filt!.PositionOfTheDefaultPresentation ) and
pos = filt!.PositionOfTheDefaultPresentation then
return M;
fi;
iso := IsomorphismOfFiltration( filt );
PushPresentationByIsomorphism( iso );
pos := PositionOfTheDefaultPresentation( M );
filt!.PositionOfTheDefaultPresentation := pos;
return M;
end );
##
InstallMethod( IsomorphismOfFiltration,
"for filtrations of homalg objects",
[ IsFiltrationOfFinitelyPresentedObjectRep ],
function( filt )
local M, degrees, l, p, gen_iso, pi, iota, filt_p_1, i, Fp_1, Mp,
d1, d0, eta0, epi, eta, emb, chi, alpha;
M := UnderlyingObject( filt );
degrees := DegreesOfFiltration( filt );
## the length of the filtration is guaranteed to be > 0
l := Length( degrees );
p := degrees[l];
## M_p -> F_p( M )
## the generalized isomorphism of the p-th graded part M_p
## onto the (filtered) object M
gen_iso := CertainMorphism( filt, p );
## end the recursion
if l = 1 then
Assert( 3, IsIsomorphism( gen_iso ) );
SetIsIsomorphism( gen_iso, true );
## transfer the known properties/attributes in both directions
UpdateObjectsByMorphism( gen_iso );
return gen_iso;
fi;
## pi: M = F_p( M ) -> M_p
## the epimorphism F_p( M ) onto M_p
pi := gen_iso ^ -1;
Assert( 3, IsEpimorphism( pi ) );
SetIsEpimorphism( pi, true );
## iota: F_{p-1}( M ) -> F_p( M )
## the embedding iota_p of F_{p-1}( M ) into F_p( M )
iota := KernelEmb( pi );
## the degrees of the induced filtration on F_{p-1}( M )
degrees := degrees{[ 1 .. l - 1 ]};
## the induced filtration on F_{p-1}( M );
filt_p_1 := rec( degrees := degrees );
for i in degrees do
filt_p_1.(String( i )) := CertainMorphism( filt, i ) / iota;
od;
filt_p_1 := HomalgAscendingFiltration( filt_p_1, IsFiltration, true );
## F_{p-1}( M ) adapted to the filtration (recursively)
Fp_1 := OnPresentationAdaptedToFiltration( filt_p_1 );
## the p-th graded part M_p
Mp := Range( pi );
## d1: K_1 -> P_0
## the embedding of the first syzygies object K_1 = K_1( M_p )
## into the free hull P_0 of M_p
d1 := SyzygiesObjectEmb( Mp );
## d0: P_0 -> M_p
## the epimorphism from the free hull P_0 (of M_p) onto M_p
d0 := CoveringEpi( Mp );
## make a copy without the morphism aid map
gen_iso := RemoveMorphismAid( gen_iso );
## eta0: P_0 -> F_p( M )
## the first lift of the identity map of M_p to a map between P_0 and F_p( M )
eta0 := DecideZero( PreCompose( d0, gen_iso ) );
## epi: P_0 + F_{p-1}( M ) -> F_p( M )
## the epimorphism from the direct sum P_0 + F_{p-1}( M ) onto F_p( M )
epi := CoproductMorphism( -eta0, iota );
## eta: K_1 -> F_{p-1}( M )
## the 1-cocycle of the extension 0 -> F_{p-1} -> F_p -> M_p -> 0
eta := DecideZero( CompleteImageSquare( d1, eta0, iota ) );
Assert( 3, IsMorphism( eta ) );
SetIsMorphism( eta, true );
## K_1 -> P_0 + F_{p-1}( M )
## the cokernel of (the embedding) K_1 -> P_0 + F_{p-1}( M ) is
## 1) isomorphic to F_p( M )
## 2) has a presentation adapted to the filtration F_p( M ) > F_{p-1}( M ) > 0
emb := ProductMorphism( d1, eta );
## P_0 + F_{p-1}( M ) -> Cokernel( K_1 -> P_0 + F_{p-1}( M ) )
## the natural epimorphism from the direct sum P_0 + F_{p-1}( M )
## onto the cokernel of K_1 -> P_0 + F_{p-1}( M ), where the latter, by construction,
## 1) is isomorphic to F_p( M ) and
## 2) has a presentation adapted to the filtration F_p( M ) > F_{p-1}( M ) > 0
chi := CokernelEpi( emb );
## the isomorphism between Cokernel( K_1 -> P_0 + F_{p-1}( M ) ) and F_p( M ),
## where the former is, by contruction, equipped with a presentation
## adapted to the filtration F_p( M ) > F_{p-1}( M ) > 0
alpha := PreDivide( chi, epi );
## freeze the computed triangular presentation
LockObjectOnCertainPresentation( Source( alpha ) );
Assert( 3, IsIsomorphism( alpha ) );
SetIsIsomorphism( alpha, true );
return alpha;
end );
##
InstallMethod( FilteredByPurity,
"for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
local filt;
filt := PurityFiltration( M );
return OnPresentationAdaptedToFiltration( filt );
end );
[ Dauer der Verarbeitung: 0.29 Sekunden
(vorverarbeitet)
]
|
