|
# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#
## Implementation stuff for homalg spectral sequences.
## we assume E collapsed to its p-axes
InstallMethod( AddTotalEmbeddingsToCollapsedToZeroSpectralSequence,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsList ],
function( E, p_range )
local E_infinity, BC, Tot, co, embeddings, n, Totn, bidegrees, tot_embs,
pq, p, q, gen_emb;
## the limit sheet of the spectral sequence
E_infinity := HighestLevelSheetInSpectralSequence( E );
## if the highest sheet is not stable issue an error
if not ( HasIsStableSheet( E_infinity ) and IsStableSheet( E_infinity ) ) and
not ( IsBound( E_infinity!.stability_table ) and IsStableSheet( E_infinity ) ) then
Error( "the highest sheet doesn't seem to be stable\n" );
fi;
#=====# begin of the core procedure #=====#
## the associated bicomplex
BC := UnderlyingBicomplex( E );
## the associated total complex
Tot := TotalComplex( BC );
if IsComplexOfFinitelyPresentedObjectsRep( Tot ) then
co := 1;
else
co := -1;
fi;
embeddings := rec( );
for n in p_range do
## the n-th total object
Totn := CertainObject( Tot, n );
## the bidegrees of total degree n
bidegrees := BidegreesOfObjectOfTotalComplex( BC, n );
## the embeddings from BC^{p,q} -> Tot^n
tot_embs := EmbeddingsInCoproductObject( Totn, bidegrees );
## get the absolute embeddings
gen_emb := E_infinity!.absolute_embeddings.(String( [ n, 0 ] ));
## create the total embeddings
if tot_embs <> fail then
if IsTransposedWRTTheAssociatedComplex( BC ) then
gen_emb := PreCompose( gen_emb, tot_embs.(String( [ 0, n ] )) );
else
gen_emb := PreCompose( gen_emb, tot_embs.(String( [ n, 0 ] )) );
fi;
fi;
## CertainMorphism( Tot, n + co ) is the minimum of what
## gen_emb needs to master the lifts it will be used for.
## We distinguish between complexes and cocomplexes
## (or between homological and cohomological spectral sequences)
gen_emb := GeneralizedMorphism( gen_emb, CertainMorphism( Tot, n + co ) );
## check assertion
Assert( 3, IsGeneralizedMonomorphism( gen_emb ) );
SetIsGeneralizedMonomorphism( gen_emb, true );
embeddings.(String( [ n, 0 ] )) := gen_emb;
od;
## now its time to enrich E
SetGeneralizedEmbeddingsInTotalObjects( E, embeddings );
return E;
end );
##
InstallMethod( AddTotalEmbeddingsToSpectralSequence,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsList ],
function( E, p_range )
local E_infinity, BC, Tot, embeddings, n, Totn, bidegrees, tot_embs,
monomorphism_aid_map, pq, pp, qq, p, q, gen_emb;
## the limit sheet of the spectral sequence
E_infinity := HighestLevelSheetInSpectralSequence( E );
## if the highest sheet is not stable issue an error
if not ( HasIsStableSheet( E_infinity ) and IsStableSheet( E_infinity ) ) and
not ( IsBound( E_infinity!.stability_table ) and IsStableSheet( E_infinity ) ) then
Error( "the highest sheet doesn't seem to be stable\n" );
fi;
#=====# begin of the core procedure #=====#
## the associated bicomplex
BC := UnderlyingBicomplex( E );
## the associated total complex
Tot := TotalComplex( BC );
embeddings := rec( );
for n in p_range do
## the n-th total object
Totn := CertainObject( Tot, n );
## the bidegrees of total degree n
bidegrees := BidegreesOfObjectOfTotalComplex( BC, n );
## the embeddings from BC^{p,q} -> Tot^n
tot_embs := EmbeddingsInCoproductObject( Totn, bidegrees );
## initialize the monomorphism aid map
monomorphism_aid_map := 0;
## in case E is a cohomological spectral sequences
if IsSpectralCosequenceOfFinitelyPresentedObjectsRep( E ) then
bidegrees := Reversed( bidegrees );
fi;
## in case E is a second spectral sequence
if IsTransposedWRTTheAssociatedComplex( BC ) then
bidegrees := Reversed( bidegrees );
fi;
## for the n-th bidegrees
for pq in bidegrees do
pp := pq[1];
qq := pq[2];
if IsTransposedWRTTheAssociatedComplex( BC ) then
p := qq;
q := pp;
else
p := pp;
q := qq;
fi;
## get the absolute embeddings
gen_emb := E_infinity!.absolute_embeddings.(String( [ p, q ] ));
## create the total embeddings
if tot_embs <> fail then
gen_emb := PreCompose( gen_emb, tot_embs.(String( [ pp, qq ] )) );
fi;
## elevate the generalized embedding to the correct height
## (i.e. to the correct subfactor of the total complex)
gen_emb := AddToMorphismAid( gen_emb, monomorphism_aid_map );
## check assertion
Assert( 3, IsGeneralizedMonomorphism( gen_emb ) );
SetIsGeneralizedMonomorphism( gen_emb, true );
embeddings.(String( [ p, q ] )) := gen_emb;
## prepare the next step
if tot_embs <> fail then
if IsHomalgMorphism( monomorphism_aid_map ) then
## for this next line we need to run through
## the bidegrees in the correct order
monomorphism_aid_map := CoproductMorphism( tot_embs.(String( [ pp, qq ] )), monomorphism_aid_map );
else
monomorphism_aid_map := tot_embs.(String( [ pp, qq ] ));
fi;
fi;
od;
od;
## now its time to enrich E
SetGeneralizedEmbeddingsInTotalObjects( E, embeddings );
return E;
end );
##
InstallMethod( AddSpectralFiltrationOfObjects,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsList, IsRecord ],
function( E, p_range, embeddings1 )
local embeddings, BC, filtration, n,
bidegrees, gen_emb1, pq, p, q, gen_emb2;
## add the embeddings in the objects of the total complex
AddTotalEmbeddingsToSpectralSequence( E, p_range );
## get them
embeddings := GeneralizedEmbeddingsInTotalObjects( E );
## the underlying bicomplex
BC := UnderlyingBicomplex( E );
## initialize an empty record to save the resulting filtration
filtration := rec( );
for n in p_range do
## the bidegrees of total degree n
bidegrees := BidegreesOfObjectOfTotalComplex( BC, n );
## the generalized embedding we want to lift with,
## i.e. of the object we want to filter
gen_emb1 := embeddings1.(String( n ));
## construct the generalized embeddings filtering
## tE^{n,0} = H^n( Tot( BC ) ) by E^{p,q}
for pq in bidegrees do
if IsTransposedWRTTheAssociatedComplex( BC ) then
p := pq[2];
q := pq[1];
else
p := pq[1];
q := pq[2];
fi;
## the generalized embedding we want to lift,
## i.e. of an object appearing in the filtration
gen_emb2 := embeddings.(String( [ p, q ] ));
## [Ba, Cor. 3.3]: gen_emb1 lifts gen_emb2
filtration.(String( [ p, q ] )) := gen_emb2 / gen_emb1; ## generalized lift
## this last line is one of the highlights in the code,
## where generalized embeddings play a decisive role
## (see the functors PostDivide and PreCompose
## in the packages for specific Abelian categories)
od;
od;
return filtration;
end );
##
InstallMethod( AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsInt, IsList ],
function( E, r, p_range )
local tE, tE_infinity, emb1, embeddings1, n, filtration;
tE := AssociatedFirstSpectralSequence( E );
## the limit sheet of the transposed spectral sequence
tE_infinity := HighestLevelSheetInSpectralSequence( tE );
## add the embeddings in the objects of the total complex
AddTotalEmbeddingsToCollapsedToZeroSpectralSequence( tE, p_range );
## get them
emb1 := GeneralizedEmbeddingsInTotalObjects( tE );
## prepare the input for AddSpectralFiltrationOfObjects
embeddings1 := rec( );
for n in p_range do
embeddings1.(String( n )) := emb1.(String( [ n, 0 ] ));
od;
## construct the generalized embeddings filtering
## tE^{n,0} = H^n( Tot( BC ) ) by E^{p,q} for all n in p_range
filtration := AddSpectralFiltrationOfObjects( E, p_range, embeddings1 );
## enrich the spectral sequence E
E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence := filtration;
end );
##
InstallMethod( AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsList ],
function( E, p_range )
AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence( E, -1, p_range );
end );
##
InstallMethod( AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsInt ],
function( E, r )
local p_range;
## the p-range of the collapsed (to its p-axes) transposed spectral sequence
p_range := ObjectDegreesOfSpectralSequence( E )[2];
AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence( E, r, p_range );
end );
##
InstallMethod( AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex ],
function( E )
AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence( E, -1 );
end );
##
InstallMethod( AddSpectralFiltrationOfTotalDefects,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex, IsList ],
function( E, p_range )
local BC, Tot, embeddings1, n, Hn, filtration;
## the underlying bicomplex
BC := UnderlyingBicomplex( E );
## the associated total complex
Tot := TotalComplex( BC );
## prepare the input for AddSpectralFiltrationOfObjects
embeddings1 := rec( );
for n in p_range do
## the n-th total (co)homology
Hn := DefectOfExactness( Tot, n );
## the n-th generalized embedding
embeddings1.(String( n )) := NaturalGeneralizedEmbedding( Hn );
od;
## construct the generalized embeddings filtering
## H^n( Tot( BC ) ) by E^{p,q} for all n in p_range
filtration := AddSpectralFiltrationOfObjects( E, p_range, embeddings1 );
## enrich the spectral sequence E
SetGeneralizedEmbeddingsInTotalDefects( E, filtration );
end );
##
InstallMethod( AddSpectralFiltrationOfTotalDefects,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex ],
function( E )
local BC, p_range;
BC := UnderlyingBicomplex( E );
p_range := TotalObjectDegreesOfBicomplex( BC );
AddSpectralFiltrationOfTotalDefects( E, p_range );
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsInt, IsInt, IsList ],
function( BC, r, a, p_range )
local tBC, tE, E;
E := HomalgSpectralSequence( BC, a );
## the transposed of the underlying bicomplex
tBC := TransposedBicomplex( BC );
## the transposed spectral sequence (w.r.t. BC),
## which we assume collapsed to its p-axes
tE := HomalgSpectralSequence( r, tBC ); ## enforce computation till the r-th sheet, even if things stabilize earlier
## enrich E with tE
E!.TransposedSpectralSequence := tE;
## filter the stable objects of the collapsed transposed spectral sequence
## with the stable objects of this spectral sequence
E!.GeneralizedEmbeddingsInStableSheetOfCollapsedTransposedSpectralSequence :=
function( )
## TODO: this is a quick hack to introduce lazyness
## (otherwise BidualizingSpectralSequence for sheaves won't work;
## we need such premature computations in our paper "Gabriel morphisms",
## all this should work as soon as Gabriel morphisms are supported in the homalg project)
AddSpectralFiltrationOfObjectsInCollapsedToZeroTransposedSpectralSequence( E, r, p_range );
end;
return E;
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsList ],
function( BC, p_range )
return SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence( BC, -1, -1, p_range );
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsInt, IsInt ],
function( BC, r, a )
local p_range;
## the p-range of the collapsed (to its p-axes) transposed spectral sequence
p_range := ObjectDegreesOfBicomplex( BC )[2];
return SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence( BC, r, a, p_range );
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence,
"for homalg bicomplexes",
[ IsHomalgBicomplex ],
function( BC )
return SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence( BC, -1, -1 );
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfTotalDefects,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsInt, IsList ],
function( BC, a, p_range )
local E;
E := HomalgSpectralSequence( BC, a );
## filter the total defects
## with the stable objects of this spectral sequence
AddSpectralFiltrationOfTotalDefects( E, p_range );
return E;
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfTotalDefects,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsList ],
function( BC, p_range )
return SpectralSequenceWithFiltrationOfTotalDefects( BC, -1, p_range );
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfTotalDefects,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsInt ],
function( BC, a )
local p_range;
p_range := TotalObjectDegreesOfBicomplex( BC );
return SpectralSequenceWithFiltrationOfTotalDefects( BC, a, p_range );
end );
##
InstallMethod( SpectralSequenceWithFiltrationOfTotalDefects,
"for homalg bicomplexes",
[ IsHomalgBicomplex ],
function( BC )
return SpectralSequenceWithFiltrationOfTotalDefects( BC, -1 );
end );
##
InstallMethod( SecondSpectralSequenceWithFiltration,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsInt, IsInt, IsList ],
function( BC, r, a, p_range )
local tBC, q_degrees, II_E;
tBC := TransposedBicomplex( BC );
q_degrees := ObjectDegreesOfBicomplex( BC )[2];
if ( IsBicomplexOfFinitelyPresentedObjectsRep( BC ) and q_degrees[Length( q_degrees )] = 0 ) or
( IsBicocomplexOfFinitelyPresentedObjectsRep( BC ) and q_degrees[1] = 0 ) then
return SpectralSequenceWithFiltrationOfCollapsedToZeroTransposedSpectralSequence( tBC, r, a, p_range );
else
II_E := SpectralSequenceWithFiltrationOfTotalDefects( tBC, a, p_range );
AssociatedFirstSpectralSequence( II_E );
return II_E;
fi;
end );
##
InstallMethod( SecondSpectralSequenceWithFiltration,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsList ],
function( BC, p_range )
return SecondSpectralSequenceWithFiltration( BC, -1, -1, p_range );
end );
##
InstallMethod( SecondSpectralSequenceWithFiltration,
"for homalg bicomplexes",
[ IsHomalgBicomplex, IsInt, IsInt ],
function( BC, r, a )
local q_degrees, p_range;
q_degrees := ObjectDegreesOfBicomplex( BC )[2];
if ( IsBicomplexOfFinitelyPresentedObjectsRep( BC ) and q_degrees[Length( q_degrees )] = 0 ) or
( IsBicocomplexOfFinitelyPresentedObjectsRep( BC ) and q_degrees[1] = 0 ) then
p_range := ObjectDegreesOfBicomplex( BC )[1];
else
p_range := TotalObjectDegreesOfBicomplex( BC );
fi;
return SecondSpectralSequenceWithFiltration( BC, r, a, p_range );
end );
##
InstallMethod( SecondSpectralSequenceWithFiltration,
"for homalg bicomplexes",
[ IsHomalgBicomplex ],
function( BC )
return SecondSpectralSequenceWithFiltration( BC, -1, -1 );
end );
##
InstallMethod( GrothendieckBicomplex,
"for homalg functors",
[ IsHomalgFunctorRep, IsHomalgFunctorRep, IsStaticFinitelyPresentedObjectRep ],
function( Functor_F, Functor_G, M )
local F, G, P, GP, CE, FCE, BC, p_degrees, FGP, HFGP, Hgen_embs, p,
natural_epis, F_natural_epis;
F := OperationOfFunctor( Functor_F );
G := OperationOfFunctor( Functor_G );
## a projective resolution of M
## (which is an injective resolution in the opposite category)
P := ShortenResolution( M );
## apply the inner functor G to the resolution P of M
GP := G( P );
## compute the Cartan-Eilenberg resolution of P
CE := Resolution( GP );
## apply the outer functor F to the Cartan-Eilenberg resolution
FCE := F( CE );
## the associated bicomplex
BC := HomalgBicomplex( FCE );
## the p-degrees
p_degrees := ObjectDegreesOfBicomplex( BC )[1];
## in case F is contravariant and left exact
## or F is covariant and right exact, then
## F( G( P ) ) is the zero-th row of first sheet
## of the collapsed (to its p-axes) first spectral sequence
FGP := F( GP );
## HFGP = H( (FG)( P ) )
## = L_*(FG)(M) (FG covariant)
## = R^*(FG)(M) (FG contravariant)
HFGP := DefectOfExactness( FGP );
## extract the natural embeddings out of H( (FG)( P ) )
Hgen_embs := rec( );
for p in p_degrees do
Hgen_embs.(String( [ p, 0 ] ) ) := NaturalGeneralizedEmbedding( CertainObject( HFGP, p ) );
od;
## enrich the bicomplex with the natural embeddings
## of H( (FG)( P ) ) into (FG)( P )
BC!.GeneralizedEmbeddingsOfHigherDerivedFunctorsOfTheComposition := Hgen_embs;
## the natural epimorphisms CE -> GP -> 0
natural_epis := CE!.NaturalEpis;
F_natural_epis := rec( );
for p in p_degrees do
F_natural_epis.(String( [ p, 0 ] )) := F( natural_epis.(String( [ p, 0 ] )) );
od;
## save the outer and the inner functor in the Grothendieck bicomplex:
BC!.OuterFunctor := Functor_F;
BC!.InnerFunctor := Functor_G;
## enrich the bicomplex with F(natural epis)
## (by this, SecondSpectralSequenceWithFiltration applied to BC
## will compute certain natural transformations needed later)
BC!.OuterFunctorOnNaturalEpis := F_natural_epis;
return BC;
end );
##
InstallMethod( BidualizingBicomplex,
"for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
local R, F, G, BC;
R := StructureObject( M );
if IsHomalgLeftObjectOrMorphismOfLeftObjects( M ) then
F := RightDualizingFunctor( R ); # Hom(-,R) for right objects
G := LeftDualizingFunctor( R ); # Hom(-,R) for left objects
else
F := LeftDualizingFunctor( R ); # Hom(-,R) for left objects
G := RightDualizingFunctor( R ); # Hom(-,R) for right objects
fi;
return GrothendieckBicomplex( F, G, M );
end );
##
InstallMethod( EnrichAssociatedFirstGrothendieckSpectralSequence,
"for homalg spectral sequences",
[ IsHomalgSpectralSequenceAssociatedToABicomplex ],
function( II_E )
local BC, Hgen_embs, I_E, I_E1, natural_transformations,
I_E2, gen_embs, p_degrees, nat_trafos, p;
## the (enriched) Grothendieck bicomplex
BC := TransposedBicomplex( UnderlyingBicomplex( II_E ) );
if IsTransposedWRTTheAssociatedComplex( BC ) then
Error( "this doesn't seem like a second spectral sequence; it is probably a first spectral sequence\n" );
fi;
## the natural embeddings of H( (FG)( P ) ) into (FG)( P )
Hgen_embs := BC!.GeneralizedEmbeddingsOfHigherDerivedFunctorsOfTheComposition;
## extract the associated first spectral sequence
I_E := AssociatedFirstSpectralSequence( II_E );
## exit if I_E is already enriched
if IsBound( I_E!.NaturalTransformations ) and
IsBound( I_E!.NaturalGeneralizedEmbeddings ) then
return;
fi;
## the first sheet of the first spectral sequence
I_E1 := CertainSheet( I_E, 1 );
## extract the natural transformations
## 0 -> F(G(P_p)) -> R^0(F)(G(P_p)) (F contravariant)
## L_0(F)(G(P_p)) -> F(G(P_p)) -> 0 (F covariant)
## out of the first sheet of the first spectral sequence
natural_transformations := I_E1!.NaturalTransformations;
## the second sheet of the first spectral sequence
I_E2 := CertainSheet( I_E, 2 );
## extract the natural embeddings out of the zero-th row
## of the second sheet of the first spectral sequence
gen_embs := I_E2!.embeddings;
## the p-degrees
p_degrees := ObjectDegreesOfBicomplex( BC )[1];
p_degrees := List( p_degrees, p -> String( [ p, 0 ]) );
## the natural transformations between the zero-th row
## of the second sheet of the first spectral sequence
## and H( (FG)( P ) )
nat_trafos := rec( );
if IsCovariantFunctor( BC!.OuterFunctor ) then
for p in p_degrees do
if IsBound( natural_transformations.(p) ) then
nat_trafos.(p) := PreCompose( gen_embs.(p), natural_transformations.(p) ) / Hgen_embs.(p); ## generalized lift
else
nat_trafos.(p) := TheZeroMorphism( Source( gen_embs.(p) ), Source( Hgen_embs.(p) ) );
fi;
Assert( 3, IsMonomorphism( nat_trafos.(p) ) );
SetIsMonomorphism( nat_trafos.(p), true );
od;
else
for p in p_degrees do
if IsBound( natural_transformations.(p) ) then
nat_trafos.(p) := gen_embs.(p) / PreCompose( Hgen_embs.(p), natural_transformations.(p) ); ## generalized lift
else
nat_trafos.(p) := TheZeroMorphism( Source( gen_embs.(p) ), Source( Hgen_embs.(p) ) );
fi;
Assert( 3, IsMonomorphism( nat_trafos.(p) ) );
SetIsMonomorphism( nat_trafos.(p), true );
od;
fi;
## enrich I_E
I_E!.NaturalTransformations := nat_trafos;
I_E!.NaturalGeneralizedEmbeddings := Hgen_embs;
end );
##
InstallMethod( GrothendieckSpectralSequence,
"for homalg functors",
[ IsHomalgFunctorRep, IsHomalgFunctorRep, IsStaticFinitelyPresentedObjectRep, IsList ],
function( Functor_F, Functor_G, M, _p_range )
local BC, p_range, II_E;
## the (enriched) Grothendieck bicomplex
BC := GrothendieckBicomplex( Functor_F, Functor_G, M );
## set the p_range
if _p_range = [ ] then
p_range := ObjectDegreesOfBicomplex( BC )[1];
else
p_range := _p_range;
fi;
## the spectral sequence II_E associated to the transposed of BC,
## also called the second spectral sequence of the bicomplex BC;
## it becomes intrinsic at the second level (w.r.t. some original data)
## (e.g. R^{-p} F R^q G => L_{p+q} FG)
##
## ... together with the filtration II_E induces
## on the objects of the collapsed (to its p-axes) first spectral sequence
## (or, equivalently, on the defects of the associated total complex)
##
## ... and since the Grothendieck bicomplex BC is enriched
## with F(natural epis) the next command will also compute
## certain natural transformations needed below
##
## the first 2 means:
## compute the first spectral sequence till the second sheet,
## even if things stabilize earlier
##
## the second 2 means:
## the second sheet of the second spectral sequence is,
## in general, the first intrinsic sheet
II_E := SecondSpectralSequenceWithFiltration( BC, 2, 2, p_range );
## astonishingly, EnrichAssociatedFirstGrothendieckSpectralSequence
## hardly causes extra computations;
## this is probably due to the caching mechanisms
## (this was observed with Purity.g)
EnrichAssociatedFirstGrothendieckSpectralSequence( II_E );
return II_E;
end );
##
InstallMethod( GrothendieckSpectralSequence,
"for homalg functors",
[ IsHomalgFunctorRep, IsHomalgFunctorRep, IsStaticFinitelyPresentedObjectRep ],
function( Functor_F, Functor_G, M )
return GrothendieckSpectralSequence( Functor_F, Functor_G, M, [ ] );
end );
##
InstallMethod( BidualizingSpectralSequence,
"for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep, IsList ],
function( M, _p_range )
local BC, p_range, II_E;
## the (enriched) Grothendieck bicomplex
BC := BidualizingBicomplex( M );
## set the p_range
if _p_range = [ ] then
p_range := ObjectDegreesOfBicomplex( BC )[1];
else
p_range := _p_range;
fi;
## the spectral sequence II_E associated to the transposed of BC,
## also called the second spectral sequence of the bicomplex BC;
## it becomes intrinsic at the second level (w.r.t. some original data)
## (e.g. R^{-p} F R^q G => L_{p+q} FG)
##
## ... together with the filtration II_E induces
## on the objects of the collapsed (to its p-axes) first spectral sequence
## (or, equivalently, on the defects of the associated total complex)
##
## ... and since the Grothendieck bicomplex BC is enriched
## with F(natural epis) the next command will also compute
## certain natural transformations needed below
##
## the first 2 means:
## compute the first spectral sequence till the second sheet,
## even if things stabilize earlier
##
## the second 2 means:
## the second sheet of the second spectral sequence is,
## in general, the first intrinsic sheet
II_E := SecondSpectralSequenceWithFiltration( BC, 2, 2, p_range );
## astonishingly, EnrichAssociatedFirstGrothendieckSpectralSequence
## hardly causes extra computations;
## this is probably due to the caching mechanisms
## (this was observed with Purity.g)
EnrichAssociatedFirstGrothendieckSpectralSequence( II_E );
return II_E;
end );
##
InstallMethod( BidualizingSpectralSequence,
"for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
return BidualizingSpectralSequence( M, [ ] );
end );
[ Dauer der Verarbeitung: 0.10 Sekunden
(vorverarbeitet)
]
|
