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## <#GAPDoc Label="A3_Purity">
## <Subsection Label="A3_Purity">
## <Heading>A3_Purity</Heading>
## This is Example B.4 in <Cite Key="BaSF"/>.
## <Example><![CDATA[
## gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
## Q[x,y,z]
## gap> A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );
## Q[x,y,z]<Dx,Dy,Dz>
## gap> nmat := HomalgMatrix( "[ \
## > 3*Dy*Dz-Dz^2+Dx+3*Dy-Dz, 3*Dy*Dz-Dz^2, \
## > Dx*Dz+Dz^2+Dz, Dx*Dz+Dz^2, \
## > Dx*Dy, 0, \
## > Dz^2-Dx+Dz, 3*Dx*Dy+Dz^2, \
## > Dx^2, 0, \
## > -Dz^2+Dx-Dz, 3*Dx^2-Dz^2, \
## > Dz^3-Dx*Dz+Dz^2, Dz^3, \
## > 2*x*Dz^2-2*x*Dx+2*x*Dz+3*Dx+3*Dz+3,2*x*Dz^2+3*Dx+3*Dz\
## > ]", 8, 2, A3 );
## <A 8 x 2 matrix over an external ring>
## gap> N := LeftPresentation( nmat );
## <A left module presented by 8 relations for 2 generators>
## gap> filt := PurityFiltration( N );
## <The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:
## 0: <A zero left module>
##
## -1: <A cyclic reflexively pure grade 1 left module presented by 1 relation for\
## a cyclic generator>
##
## -2: <A cyclic reflexively pure grade 2 left module presented by 2 relations fo\
## r a cyclic generator>
##
## -3: <A cyclic reflexively pure grade 3 left module presented by 3 relations fo\
## r a cyclic generator>
## of
## <A non-pure grade 1 left module presented by 8 relations for 2 generators>>
## gap> II_E := SpectralSequence( filt );
## <A stable homological spectral sequence with sheets at levels
## [ 0 .. 2 ] each consisting of left modules at bidegrees [ -4 .. 0 ]x
## [ 0 .. 3 ]>
## gap> Display( II_E );
## The associated transposed spectral sequence:
##
## a homological spectral sequence at bidegrees
## [ [ 0 .. 3 ], [ -4 .. 0 ] ]
## ---------
## Level 0:
##
## * * * *
## . * * *
## . . * *
## . . . *
## . . . *
## ---------
## Level 1:
##
## * * * *
## . . . .
## . . . .
## . . . .
## . . . .
## ---------
## Level 2:
##
## s . . .
## . . . .
## . . . .
## . . . .
## . . . .
##
## Now the spectral sequence of the bicomplex:
##
## a homological spectral sequence at bidegrees
## [ [ -4 .. 0 ], [ 0 .. 3 ] ]
## ---------
## Level 0:
##
## * * * * *
## . . * * *
## . . . * *
## . . . . *
## ---------
## Level 1:
##
## * * * * *
## . . * * *
## . . . * *
## . . . . .
## ---------
## Level 2:
##
## . s . . .
## . . s . .
## . . . s .
## . . . . .
## gap> m := IsomorphismOfFiltration( filt );
## <A non-zero isomorphism of left modules>
## gap> IsIdenticalObj( Range( m ), N );
## true
## gap> Source( m );
## <A left module presented by 6 relations for 3 generators (locked)>
## gap> Display( last );
## Dx,1/3,1/216*x,
## 0, Dy, -1/144,
## 0, Dx, 1/48,
## 0, 0, Dz,
## 0, 0, Dy,
## 0, 0, Dx
##
## Cokernel of the map
##
## R^(1x6) --> R^(1x3), ( for R := Q[x,y,z]<Dx,Dy,Dz> )
##
## currently represented by the above matrix
## gap> Display( filt );
## Degree 0:
##
## 0
## ----------
## Degree -1:
##
## Q[x,y,z]<Dx,Dy,Dz>/< Dx >
## ----------
## Degree -2:
##
## Q[x,y,z]<Dx,Dy,Dz>/< Dy, Dx >
## ----------
## Degree -3:
##
## Q[x,y,z]<Dx,Dy,Dz>/< Dz, Dy, Dx >
## gap> Display( m );
## 1, 1,
## 3*Dz+3, 3*Dz,
## 144*Dz^2-144*Dx+144*Dz,144*Dz^2
##
## the map is currently represented by the above 3 x 2 matrix
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
ReadPackage( "ExamplesForHomalg", "examples/Coupling.g" );
filt := PurityFiltration( N );
II_E := SpectralSequence( filt );
m := IsomorphismOfFiltration( filt );
#Display( StringTime( homalgTime( A3 ) ) );
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