# ExamplesForHomalg, single 5
#
# 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("examplesforhomalg05.tst");
# doc/../examples/TorExt.g:5-122
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> P := Resolution( W );
<A right acyclic complex containing 3 morphisms of left modules at degrees
[ 0 .. 3 ]>
gap> GP := Hom( P );
<A cocomplex containing 3 morphisms of right modules at degrees [ 0 .. 3 ]>
gap> FGP := GP * P;
<A cocomplex containing 3 morphisms of left complexes at degrees [ 0 .. 3 ]>
gap> BC := HomalgBicomplex( FGP );
<A bicocomplex containing left modules at bidegrees [ 0 .. 3 ]x[ -3 .. 0 ]>
gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1];
[ 0 .. 3 ]
gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );
<A stable cohomological 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 cohomological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -3 .. 0 ] ]
---------
Level 0:
* * * *
* * * *
* * * *
* * * *
---------
Level 1:
* * * *
. . . .
. . . .
. . . .
---------
Level 2:
s s s s
. . . .
. . . .
. . . .
Now the spectral sequence of the bicomplex:
a cohomological spectral sequence at bidegrees
[ [ -3 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:
* * * *
* * * *
* * * *
* * * *
---------
Level 1:
* * * *
* * * *
* * * *
* * * *
---------
Level 2:
* * s s
* * * *
. * * *
. . . *
---------
Level 3:
* s s s
. s s s
. . s *
. . . s
---------
Level 4:
s s s s
. s s s
. . s s
. . . s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<A descending filtration with degrees [ -3 .. 0 ] and graded parts:
-3: <A non-zero cyclic torsion left module presented by yet unknown relations \
for a cyclic generator>
-2: <A non-zero left module presented by 15 relations for 6 generators>
-1: <A non-zero left module presented by 27 relations for 13 generators>
0: <A non-zero left module presented by 13 relations for 10 generators>
of
<A left module presented by yet unknown relations for 31 generators>>
gap> ByASmallerPresentation( filt );
<A descending filtration with degrees [ -3 .. 0 ] and graded parts:
-3: <A non-zero cyclic torsion left module presented by 3 relations for a cycl\
ic generator>
-2: <A non-zero left module presented by 11 relations for 4 generators>
-1: <A non-zero left module presented by 23 relations for 9 generators>
0: <A non-zero left module presented by 11 relations for 10 generators>
of
<A non-zero left module presented by 24 relations for 12 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
#
gap> STOP_TEST("examplesforhomalg05.tst", 1);
