# ExamplesForHomalg, single 6
#
# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!
#
# This file has been generated by AutoDoc. It contains examples extracted from
# the package documentation. Each example is preceded by a comment which gives
# the name of a GAPDoc XML file and a line range from which the example were
# taken. Note that the XML file in turn may have been generated by AutoDoc
# from some other input.
#
gap> START_TEST("examplesforhomalg06.tst");
# doc/../examples/CodegreeOfPurity.g:5-190
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
#
gap> STOP_TEST("examplesforhomalg06.tst", 1);
