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## <#GAPDoc Label="CodegreeOfPurity">
## <Subsection Label="CodegreeOfPurity">
## <Heading>CodegreeOfPurity</Heading>
## This is Example B.7 in <Cite Key="BaSF"/>.
## <Example><![CDATA[
## gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
## Q[x,y,z]
## gap> vmat := HomalgMatrix( "[ \
## > 0, 0, x,-z, \
## > x*z,z^2,y,0, \
## > x^2,x*z,0,y \
## > ]", 3, 4, Qxyz );
## <A 3 x 4 matrix over an external ring>
## gap> V := LeftPresentation( vmat );
## <A non-torsion left module presented by 3 relations for 4 generators>
## gap> wmat := HomalgMatrix( "[ \
## > 0, 0, x,-y, \
## > x*y,y*z,z,0, \
## > x^2,x*z,0,z \
## > ]", 3, 4, Qxyz );
## <A 3 x 4 matrix over an external ring>
## gap> W := LeftPresentation( wmat );
## <A non-torsion left module presented by 3 relations for 4 generators>
## gap> Rank( V );
## 2
## gap> Rank( W );
## 2
## gap> ProjectiveDimension( V );
## 2
## gap> ProjectiveDimension( W );
## 2
## gap> DegreeOfTorsionFreeness( V );
## 1
## gap> DegreeOfTorsionFreeness( W );
## 1
## gap> CodegreeOfPurity( V );
## [ 2 ]
## gap> CodegreeOfPurity( W );
## [ 1, 1 ]
## gap> filtV := PurityFiltration( V );
## <The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts:
##
## 0: <A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 ge\
## nerators>
## -1: <A zero left module>
## -2: <A zero left module>
## of
## <A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 gener\
## ators>>
## gap> filtW := PurityFiltration( W );
## <The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts:
##
## 0: <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\
## generators>
## -1: <A zero left module>
## -2: <A zero left module>
## of
## <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4 ge\
## nerators>>
## gap> II_EV := SpectralSequence( filtV );
## <A stable homological spectral sequence with sheets at levels
## [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
## [ 0 .. 2 ]>
## gap> Display( II_EV );
## The associated transposed spectral sequence:
##
## a homological spectral sequence at bidegrees
## [ [ 0 .. 2 ], [ -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 .. 2 ] ]
## ---------
## Level 0:
##
## * * * *
## * * * *
## . * * *
## ---------
## Level 1:
##
## * * * *
## * * * *
## . . * *
## ---------
## Level 2:
##
## * . . .
## * . . .
## . . * *
## ---------
## Level 3:
##
## * . . .
## . . . .
## . . . *
## ---------
## Level 4:
##
## . . . .
## . . . .
## . . . s
## gap> II_EW := SpectralSequence( filtW );
## <A stable homological spectral sequence with sheets at levels
## [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
## [ 0 .. 2 ]>
## gap> Display( II_EW );
## The associated transposed spectral sequence:
##
## a homological spectral sequence at bidegrees
## [ [ 0 .. 2 ], [ -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 .. 2 ] ]
## ---------
## Level 0:
##
## * * * *
## . * * *
## . . * *
## ---------
## Level 1:
##
## * * * *
## . * * *
## . . . *
## ---------
## Level 2:
##
## * . . .
## . * . .
## . . . *
## ---------
## Level 3:
##
## * . . .
## . . . .
## . . . *
## ---------
## Level 4:
##
## . . . .
## . . . .
## . . . s
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
LoadPackage( "RingsForHomalg" );
Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
vmat := HomalgMatrix( "[ \
x,z,0,1, 0, \
0,0,y,-z,0, \
0,0,x,0, -z, \
0,0,0,x, -y \
]", 4, 5, Qxyz );
wmat := HomalgMatrix( "[ \
x,z,1,0, 0, \
0,0,y,-z,0, \
0,0,x,0, -z, \
0,0,0,x, -y \
]", 4, 5, Qxyz );
LoadPackage( "Modules" );
V := LeftPresentation( vmat );
W := LeftPresentation( wmat );
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