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## <#GAPDoc Label="Purity">
## <Subsection Label="Purity">
## <Heading>Purity</Heading>
## This is Example B.3 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> filt := PurityFiltration( W );
## <The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:
##
## 0: <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\
## generators>
##
## -1: <A codegree-1-pure grade 1 left module presented by 4 relations for 3 gene\
## rators>
##
## -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 rank 2 left module presented by 6 relations for 5 generators>>
## gap> W;
## <A non-pure rank 2 left module presented by 6 relations for 5 generators>
## gap> II_E := SpectralSequence( filt );
## <A stable homological 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 homological spectral sequence at bidegrees
## [ [ 0 .. 3 ], [ -3 .. 0 ] ]
## ---------
## Level 0:
##
## * * * *
## * * * *
## . * * *
## . . * *
## ---------
## Level 1:
##
## * * * *
## . . . .
## . . . .
## . . . .
## ---------
## Level 2:
##
## s . . .
## . . . .
## . . . .
## . . . .
##
## Now the spectral sequence of the bicomplex:
##
## a homological spectral sequence at bidegrees
## [ [ -3 .. 0 ], [ 0 .. 3 ] ]
## ---------
## Level 0:
##
## * * * *
## * * * *
## . * * *
## . . * *
## ---------
## Level 1:
##
## * * * *
## * * * *
## . * * *
## . . . *
## ---------
## Level 2:
##
## s . . .
## * s . .
## . * * .
## . . . *
## ---------
## Level 3:
##
## s . . .
## * s . .
## . . s .
## . . . *
## ---------
## Level 4:
##
## s . . .
## . s . .
## . . s .
## . . . s
##
## gap> m := IsomorphismOfFiltration( filt );
## <A non-zero isomorphism of left modules>
## gap> IsIdenticalObj( Range( m ), W );
## true
## gap> Source( m );
## <A left module presented by 12 relations for 9 generators (locked)>
## gap> Display( last );
## 0, 0,x, -y,0,1, 0, 0, 0,
## x*y,0,-z,0, 0,0, 0, 0, 0,
## x^2,0,0, -z,1,0, 0, 0, 0,
## 0, 0,0, 0, y,-z,0, 0, 0,
## 0, 0,0, 0, 0,x, -y, -1, 0,
## 0, 0,0, 0, x,0, -z, 0, -1,
## 0, 0,0, 0, 0,-y,x^2-1,0, 0,
## 0, 0,0, 0, 0,0, 0, z, 0,
## 0, 0,0, 0, 0,0, 0, y-1,0,
## 0, 0,0, 0, 0,0, 0, 0, z,
## 0, 0,0, 0, 0,0, 0, 0, y,
## 0, 0,0, 0, 0,0, 0, 0, x
##
## Cokernel of the map
##
## Q[x,y,z]^(1x12) --> Q[x,y,z]^(1x9),
##
## currently represented by the above matrix
## gap> Display( filt );
## Degree 0:
##
## 0, 0,x, -y,
## x*y,0,-z,0,
## x^2,0,0, -z
##
## Cokernel of the map
##
## Q[x,y,z]^(1x3) --> Q[x,y,z]^(1x4),
##
## currently represented by the above matrix
## ----------
## Degree -1:
##
## y,-z,0,
## 0,x, -y,
## x,0, -z,
## 0,-y,x^2-1
##
## Cokernel of the map
##
## Q[x,y,z]^(1x4) --> Q[x,y,z]^(1x3),
##
## currently represented by the above matrix
## ----------
## Degree -2:
##
## Q[x,y,z]/< z, y-1 >
## ----------
## Degree -3:
##
## Q[x,y,z]/< z, y, x >
## gap> Display( m );
## 1, 0, 0, 0, 0,
## 0, 1, 0, 0, 0,
## 0, -y, -1, 0, 0,
## 0, -x, 0, -1, 0,
## -x^2,-x*z, 0, -z, 0,
## 0, 0, x, -y, 0,
## 0, 0, 0, 0, -1,
## 0, 0, x^2,-x*y,y,
## -x^3,-x^2*z,0, -x*z,z
##
## the map is currently represented by the above 9 x 5 matrix
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
ReadPackage( "ExamplesForHomalg", "examples/ReducedBasisOfModule.g" );
filt := PurityFiltration( W );
II_E := SpectralSequence( filt );
m := IsomorphismOfFiltration( filt );
#Display( StringTime( homalgTime( Qxyz ) ) );
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