Quelle examplesforhomalg02.tst Sprache: unbekannt

# ExamplesForHomalg, single 2
#
# 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("examplesforhomalg02.tst");

# doc/../examples/Purity.g:5-178
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> filt := PurityFiltration( W );
<The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:

0:   <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\
generators>

-1:   <A codegree-1-pure grade 1 left module presented by 4 relations for 3 gene\
rators>

-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 rank 2 left module presented by 6 relations for 5 generators>>
gap> W;
<A non-pure rank 2 left module presented by 6 relations for 5 generators>
gap> II_E := SpectralSequence( filt );
<A stable homological 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 homological spectral sequence at bidegrees
[ [ 0 .. 3 ], [ -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 .. 3 ] ]
---------
Level 0:

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

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

s . . .
* s . .
. * * .
. . . *
---------
Level 3:

s . . .
* s . .
. . s .
. . . *
---------
Level 4:

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

gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
gap> IsIdenticalObj( Range( m ), W );
true
gap> Source( m );
<A left module presented by 12 relations for 9 generators (locked)>
gap> Display( last );
0,  0,x, -y,0,1, 0,    0,  0,
x*y,0,-z,0, 0,0, 0,    0,  0,
x^2,0,0, -z,1,0, 0,    0,  0,
0,  0,0, 0, y,-z,0,    0,  0,
0,  0,0, 0, 0,x, -y,   -1, 0,
0,  0,0, 0, x,0, -z,   0,  -1,
0,  0,0, 0, 0,-y,x^2-1,0,  0,
0,  0,0, 0, 0,0, 0,    z,  0,
0,  0,0, 0, 0,0, 0,    y-1,0,
0,  0,0, 0, 0,0, 0,    0,  z,
0,  0,0, 0, 0,0, 0,    0,  y,
0,  0,0, 0, 0,0, 0,    0,  x

Cokernel of the map

Q[x,y,z]^(1x12) --> Q[x,y,z]^(1x9),

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

0,  0,x, -y,
x*y,0,-z,0,
x^2,0,0, -z

Cokernel of the map

Q[x,y,z]^(1x3) --> Q[x,y,z]^(1x4),

currently represented by the above matrix
----------
Degree -1:

y,-z,0,
0,x, -y,
x,0, -z,
0,-y,x^2-1

Cokernel of the map

Q[x,y,z]^(1x4) --> Q[x,y,z]^(1x3),

currently represented by the above matrix
----------
Degree -2:

Q[x,y,z]/< z, y-1 >
----------
Degree -3:

Q[x,y,z]/< z, y, x >
gap> Display( m );
1,   0,     0,  0,   0,
0,   1,     0,  0,   0,
0,   -y,    -1, 0,   0,
0,   -x,    0,  -1,  0,
-x^2,-x*z,  0,  -z,  0,
0,   0,     x,  -y,  0,
0,   0,     0,  0,   -1,
0,   0,     x^2,-x*y,y,
-x^3,-x^2*z,0,  -x*z,z

the map is currently represented by the above 9 x 5 matrix

#
gap> STOP_TEST("examplesforhomalg02.tst", 1);

Quelle examplesforhomalg02.tst Sprache: unbekannt

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