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## <#GAPDoc Label="TorExt">
## <Subsection Label="TorExt">
## <Heading>TorExt</Heading>
## This is Example B.6 in <Cite Key="BaSF"/>.
## <Example><![CDATA[
## gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
## Q[x,y,z]
## gap> wmat := HomalgMatrix( "[ \
## > x*y, y*z, z, 0, 0, \
## > x^3*z,x^2*z^2,0, x*z^2, -z^2, \
## > x^4, x^3*z, 0, x^2*z, -x*z, \
## > 0, 0, x*y, -y^2, x^2-1,\
## > 0, 0, x^2*z, -x*y*z, y*z, \
## > 0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \
## > ]", 6, 5, Qxyz );
## <A 6 x 5 matrix over an external ring>
## gap> W := LeftPresentation( wmat );
## <A left module presented by 6 relations for 5 generators>
## gap> P := Resolution( W );
## <A right acyclic complex containing 3 morphisms of left modules at degrees
## [ 0 .. 3 ]>
## gap> GP := Hom( P );
## <A cocomplex containing 3 morphisms of right modules at degrees [ 0 .. 3 ]>
## gap> FGP := GP * P;
## <A cocomplex containing 3 morphisms of left complexes at degrees [ 0 .. 3 ]>
## gap> BC := HomalgBicomplex( FGP );
## <A bicocomplex containing left modules at bidegrees [ 0 .. 3 ]x[ -3 .. 0 ]>
## gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1];
## [ 0 .. 3 ]
## gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );
## <A stable cohomological spectral sequence with sheets at levels
## [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
## [ 0 .. 3 ]>
## gap> Display( II_E );
## The associated transposed spectral sequence:
##
## a cohomological spectral sequence at bidegrees
## [ [ 0 .. 3 ], [ -3 .. 0 ] ]
## ---------
## Level 0:
##
## * * * *
## * * * *
## * * * *
## * * * *
## ---------
## Level 1:
##
## * * * *
## . . . .
## . . . .
## . . . .
## ---------
## Level 2:
##
## s s s s
## . . . .
## . . . .
## . . . .
##
## Now the spectral sequence of the bicomplex:
##
## a cohomological spectral sequence at bidegrees
## [ [ -3 .. 0 ], [ 0 .. 3 ] ]
## ---------
## Level 0:
##
## * * * *
## * * * *
## * * * *
## * * * *
## ---------
## Level 1:
##
## * * * *
## * * * *
## * * * *
## * * * *
## ---------
## Level 2:
##
## * * s s
## * * * *
## . * * *
## . . . *
## ---------
## Level 3:
##
## * s s s
## . s s s
## . . s *
## . . . s
## ---------
## Level 4:
##
## s s s s
## . s s s
## . . s s
## . . . s
## gap> filt := FiltrationBySpectralSequence( II_E, 0 );
## <A descending filtration with degrees [ -3 .. 0 ] and graded parts:
##
## -3: <A non-zero cyclic torsion left module presented by yet unknown relations \
## for a cyclic generator>
## -2: <A non-zero left module presented by 15 relations for 6 generators>
## -1: <A non-zero left module presented by 27 relations for 13 generators>
## 0: <A non-zero left module presented by 13 relations for 10 generators>
## of
## <A left module presented by yet unknown relations for 31 generators>>
## gap> ByASmallerPresentation( filt );
## <A descending filtration with degrees [ -3 .. 0 ] and graded parts:
##
## -3: <A non-zero cyclic torsion left module presented by 3 relations for a cycl\
## ic generator>
## -2: <A non-zero left module presented by 11 relations for 4 generators>
## -1: <A non-zero left module presented by 23 relations for 9 generators>
## 0: <A non-zero left module presented by 11 relations for 10 generators>
## of
## <A non-zero left module presented by 24 relations for 12 generators>>
## gap> m := IsomorphismOfFiltration( filt );
## <A non-zero isomorphism of left modules>
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
ReadPackage( "ExamplesForHomalg", "examples/ReducedBasisOfModule.g" );
## compute a free resolution of W
P := Resolution( W );
## apply the inner functor G := Hom(-,R) to the resolution
GP := Hom( P );
## tensor with P again
FGP := GP * P;
## the bicomplex associated to FGP
BC := HomalgBicomplex( FGP );
p_degrees := ObjectDegreesOfBicomplex( BC )[1];
## the second spectral sequence together with
## the collapsed first spectral sequence
II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );
filt := FiltrationBySpectralSequence( II_E );
ByASmallerPresentation( filt );
m := IsomorphismOfFiltration( filt );
#Display( StringTime( homalgTime( Qxyz ) ) );
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