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## <#GAPDoc Label="Purity">
## <Section Label="Purity">
## <Heading>Purity</Heading>
## This corresponds to Example B.3 in <Cite Key="BaSF"/>.
## <Example><![CDATA[
## gap> zz := HomalgRingOfIntegers( );
## Z
## gap> imat := HomalgMatrix( "[ \
## > 262, -33, 75, -40, \
## > 682, -86, 196, -104, \
## > 1186, -151, 341, -180, \
## > -1932, 248, -556, 292, \
## > 1018, -127, 293, -156 \
## > ]", 5, 4, zz );
## <A 5 x 4 matrix over an internal ring>
## gap> M := LeftPresentation( imat );
## <A left module presented by 5 relations for 4 generators>
## gap> filt := PurityFiltration( M );
## <The ascending purity filtration with degrees [ -1 .. 0 ] and graded parts:
## 0: <A free left module of rank 1 on a free generator>
##
## -1: <A non-zero torsion left module presented by 2 relations for 2 generators>
## of
## <A non-pure rank 1 left module presented by 2 relations for 3 generators>>
## gap> M;
## <A non-pure rank 1 left module presented by 2 relations for 3 generators>
## gap> II_E := SpectralSequence( filt );
## <A stable homological spectral sequence with sheets at levels
## [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
## [ 0 .. 1 ]>
## gap> Display( II_E );
## The associated transposed spectral sequence:
##
## a homological spectral sequence at bidegrees
## [ [ 0 .. 1 ], [ -1 .. 0 ] ]
## ---------
## Level 0:
##
## * *
## * *
## ---------
## Level 1:
##
## * *
## . .
## ---------
## Level 2:
##
## s .
## . .
##
## Now the spectral sequence of the bicomplex:
##
## a homological spectral sequence at bidegrees
## [ [ -1 .. 0 ], [ 0 .. 1 ] ]
## ---------
## Level 0:
##
## * *
## * *
## ---------
## Level 1:
##
## * *
## . s
## ---------
## Level 2:
##
## s .
## . s
## gap> m := IsomorphismOfFiltration( filt );
## <A non-zero isomorphism of left modules>
## gap> IsIdenticalObj( Range( m ), M );
## true
## gap> Source( m );
## <A non-torsion left module presented by 2 relations for 3 generators (locked)>
## gap> Display( last );
## [ [ 0, 2, 0 ],
## [ 0, 0, 12 ] ]
##
## Cokernel of the map
##
## Z^(1x2) --> Z^(1x3),
##
## currently represented by the above matrix
## gap> Display( filt );
## Degree 0:
##
## Z^(1 x 1)
## ----------
## Degree -1:
##
## Z/< 2 > + Z/< 12 >
## ]]></Example>
## </Section>
## <#/GAPDoc>
Read( "homalg.g" );
filt := PurityFiltration( M );
II_E := SpectralSequence( filt );
m := IsomorphismOfFiltration( filt );
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