Quelle examplesforhomalg03.tst Sprache: unbekannt

# ExamplesForHomalg, single 3
#
# 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("examplesforhomalg03.tst");

# doc/../examples/A3_Purity.g:5-137
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
Q[x,y,z]
gap> A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );
Q[x,y,z]<Dx,Dy,Dz>
gap> nmat := HomalgMatrix( "[ \
> 3*Dy*Dz-Dz^2+Dx+3*Dy-Dz,           3*Dy*Dz-Dz^2,     \
> Dx*Dz+Dz^2+Dz,                     Dx*Dz+Dz^2,       \
> Dx*Dy,                             0,                \
> Dz^2-Dx+Dz,                        3*Dx*Dy+Dz^2,     \
> Dx^2,                              0,                \
> -Dz^2+Dx-Dz,                       3*Dx^2-Dz^2,      \
> Dz^3-Dx*Dz+Dz^2,                   Dz^3,             \
> 2*x*Dz^2-2*x*Dx+2*x*Dz+3*Dx+3*Dz+3,2*x*Dz^2+3*Dx+3*Dz\
> ]", 8, 2, A3 );
<A 8 x 2 matrix over an external ring>
gap> N := LeftPresentation( nmat );
<A left module presented by 8 relations for 2 generators>
gap> filt := PurityFiltration( N );
<The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:
   0:   <A zero left module>

-1:   <A cyclic reflexively pure grade 1 left module presented by 1 relation for\
a cyclic generator>

-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 grade 1 left module presented by 8 relations for 2 generators>>
gap> II_E := SpectralSequence( filt );
<A stable homological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -4 .. 0 ]x
[ 0 .. 3 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a homological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -4 .. 0 ] ]
---------
Level 0:

* * * *
. * * *
. . * *
. . . *
. . . *
---------
Level 1:

* * * *
. . . .
. . . .
. . . .
. . . .
---------
Level 2:

s . . .
. . . .
. . . .
. . . .
. . . .

Now the spectral sequence of the bicomplex:

a homological spectral sequence at bidegrees
[ [ -4 .. 0 ], [ 0 .. 3 ] ]
---------
Level 0:

* * * * *
. . * * *
. . . * *
. . . . *
---------
Level 1:

* * * * *
. . * * *
. . . * *
. . . . .
---------
Level 2:

. s . . .
. . s . .
. . . s .
. . . . .
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
gap> IsIdenticalObj( Range( m ), N );
true
gap> Source( m );
<A left module presented by 6 relations for 3 generators (locked)>
gap> Display( last );
Dx,1/3,1/216*x,
0, Dy, -1/144,
0, Dx, 1/48,
0, 0,  Dz,
0, 0,  Dy,
0, 0,  Dx

Cokernel of the map

R^(1x6) --> R^(1x3), ( for R := Q[x,y,z]<Dx,Dy,Dz> )

currently represented by the above matrix
gap> Display( filt );
Degree 0:

0
----------
Degree -1:

Q[x,y,z]<Dx,Dy,Dz>/< Dx >
----------
Degree -2:

Q[x,y,z]<Dx,Dy,Dz>/< Dy, Dx >
----------
Degree -3:

Q[x,y,z]<Dx,Dy,Dz>/< Dz, Dy, Dx >
gap> Display( m );
1,                     1,
3*Dz+3,                3*Dz,
144*Dz^2-144*Dx+144*Dz,144*Dz^2

the map is currently represented by the above 3 x 2 matrix

#
gap> STOP_TEST("examplesforhomalg03.tst", 1);

Quelle examplesforhomalg03.tst Sprache: unbekannt

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