Quelle chap3.html Sprache: HTML

products/sources/formale Sprachen/GAP/pkg/examplesforhomalg/doc/chap3.html

<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ExamplesForHomalg) - Chapter 3: Examples</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap3" onload="jscontent()">

<div class="chlinktop">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap2.html">[Previous Chapter]</a> <a href="chapBib.html">[Next Chapter]</a> </div>

<a href="chap3_mj.html">[MathJax on]</a>
<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap3.html#X7A489A5D79DA9E5C">3 Examples</a>
<div class="ContSect"> <a href="chap3.html#X7AD67ACA80E44216">3.1 Spectral Filtrations</a>

<div class="ContSSBlock">
 <a href="chap3.html#X7BB9DE017ECE6E86">3.1-1 ExtExt</a>

 <a href="chap3.html#X7EE63228803A04F1">3.1-2 Purity</a>

 <a href="chap3.html#X7816D6ED815ED641">3.1-3 A3_Purity</a>

 <a href="chap3.html#X812EF8147AE16E72">3.1-4 TorExt-Grothendieck</a>

 <a href="chap3.html#X784BC2567875830B">3.1-5 TorExt</a>

 <a href="chap3.html#X8021C33D85444081">3.1-6 CodegreeOfPurity</a>

 <a href="chap3.html#X791E21F47805048A">3.1-7 HomHom</a>

</div></div>
<div class="ContSect"> <a href="chap3.html#X85CF19B87D1C375F">3.2 Commutative Algebra</a>

<div class="ContSSBlock">
 <a href="chap3.html#X781B1C0C80529B09">3.2-1 Eliminate</a>

</div></div>
</div>

<h3>3 Examples</h3>

<a id="X7AD67ACA80E44216" name="X7AD67ACA80E44216"></a>

<h4>3.1 Spectral Filtrations</h4>

<a id="X7BB9DE017ECE6E86" name="X7BB9DE017ECE6E86"></a>

<h5>3.1-1 ExtExt</h5>

This is Example B.2 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
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> Y := Hom( Qxyz, W );
<A right module on 5 generators satisfying yet unknown relations>
gap> SetInfoLevel( InfoWarning, 0 );
gap> F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, Y, "TensorY" );
<The functor TensorY for f.p. modules and their maps over computable rings>
gap> SetInfoLevel( InfoWarning, 1 );
gap> G := LeftDualizingFunctor( Qxyz );;
gap> II_E := GrothendieckSpectralSequence( F, G, W );
<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 s s 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
* s s s
. . s *
. . . *
---------
Level 4:

s s s s
. s s s
. . s s
. . . s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<An ascending filtration with degrees [ -3 .. 0 ] and graded parts:

0: <A non-zero left module presented by yet unknown relations for 23 generator\
s>
 -1: <A non-zero left module presented by 37 relations for 22 generators>
 -2: <A non-zero left module presented by 31 relations for 10 generators>
 -3: <A non-zero left module presented by 33 relations for 5 generators>
of
<A non-zero left module presented by 102 relations for 37 generators>>
gap> ByASmallerPresentation( filt );
<An ascending filtration with degrees [ -3 .. 0 ] and graded parts:
 0: <A non-zero left module presented by 26 relations for 16 generators>
 -1: <A non-zero left module presented by 30 relations for 14 generators>
 -2: <A non-zero left module presented by 18 relations for 7 generators>
 -3: <A non-zero left module presented by 12 relations for 4 generators>
of
<A non-zero left module presented by 48 relations for 20 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

<a id="X7EE63228803A04F1" name="X7EE63228803A04F1"></a>

<h5>3.1-2 Purity</h5>

This is Example B.3 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
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
</pre></div>

<a id="X7816D6ED815ED641" name="X7816D6ED815ED641"></a>

<h5>3.1-3 A3_Purity</h5>

This is Example B.4 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
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
</pre></div>

<a id="X812EF8147AE16E72" name="X812EF8147AE16E72"></a>

<h5>3.1-4 TorExt-Grothendieck</h5>

This is Example B.5 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
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> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, W, "TensorW" );
<The functor TensorW for f.p. modules and their maps over computable rings>
gap> G := LeftDualizingFunctor( Qxyz );;
gap> II_E := GrothendieckSpectralSequence( F, G, W );
<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 17 relations for 6 generators>
 -1: <A non-zero left module presented by 27 relations for 12 generators>
 0: <A non-zero left module presented by 13 relations for 10 generators>
of
<A left module presented by yet unknown relations for 49 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 12 relations for 4 generators>
 -1: <A non-zero left module presented by 21 relations for 8 generators>
 0: <A non-zero left module presented by 11 relations for 10 generators>
of
<A non-zero left module presented by 27 relations for 14 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

<a id="X784BC2567875830B" name="X784BC2567875830B"></a>

<h5>3.1-5 TorExt</h5>

This is Example B.6 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
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>
</pre></div>

<a id="X8021C33D85444081" name="X8021C33D85444081"></a>

<h5>3.1-6 CodegreeOfPurity</h5>

This is Example B.7 in <a href="chapBib.html#biBBaSF">[Bar09]</a>.

<div class="example"><pre>
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
</pre></div>

<a id="X791E21F47805048A" name="X791E21F47805048A"></a>

<h5>3.1-7 HomHom</h5>

This corresponds to the example of Section 2 in <a href="chapBib.html#biBBREACA">[BR06]</a>.

<div class="example"><pre>
gap> R := HomalgRingOfIntegersInExternalGAP( ) / 2^8;
Z/( 256 )
gap> Display( R );
<A residue class ring>
gap> M := LeftPresentation( [ 2^5 ], R );
<A cyclic left module presented by 1 relation for a cyclic generator>
gap> Display( M );
Z/( 256 )/< |[ 32 ]| >
gap> M;
<A cyclic left module presented by 1 relation for a cyclic generator>
gap> _M := LeftPresentation( [ 2^3 ], R );
<A cyclic left module presented by 1 relation for a cyclic generator>
gap> Display( _M );
Z/( 256 )/< |[ 8 ]| >
gap> _M;
<A cyclic left module presented by 1 relation for a cyclic generator>
gap> alpha2 := HomalgMap( [ 1 ], M, _M );
<A "homomorphism" of left modules>
gap> IsMorphism( alpha2 );
true
gap> alpha2;
<A homomorphism of left modules>
gap> Display( alpha2 );
[ [ 1 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
gap> M_ := Kernel( alpha2 );
<A cyclic left module presented by yet unknown relations for a cyclic generato\
r>
gap> alpha1 := KernelEmb( alpha2 );
<A monomorphism of left modules>
gap> seq := HomalgComplex( alpha2 );
<An acyclic complex containing a single morphism of left modules at degrees
[ 0 .. 1 ]>
gap> Add( seq, alpha1 );
gap> seq;
<A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]>
gap> IsShortExactSequence( seq );
true
gap> seq;
<A short exact sequence containing 2 morphisms of left modules at degrees
[ 0 .. 2 ]>
gap> Display( seq );
-------------------------
at homology degree: 2
Z/( 256 )/< |[ 4 ]| >
-------------------------
[ [ 8 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 1
Z/( 256 )/< |[ 32 ]| >
-------------------------
[ [ 1 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 0
Z/( 256 )/< |[ 8 ]| >
-------------------------
gap> K := LeftPresentation( [ 2^7 ], R );
<A cyclic left module presented by 1 relation for a cyclic generator>
gap> L := RightPresentation( [ 2^4 ], R );
<A cyclic right module on a cyclic generator satisfying 1 relation>
gap> triangle := LHomHom( 4, seq, K, L, "t" );
<An exact triangle containing 3 morphisms of left complexes at degrees
[ 1, 2, 3, 1 ]>
gap> lehs := LongSequence( triangle );
<A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]>
gap> ByASmallerPresentation( lehs );
<A non-zero sequence containing 14 morphisms of left modules at degrees
[ 0 .. 14 ]>
gap> IsExactSequence( lehs );
false
gap> lehs;
<A non-zero left acyclic complex containing
14 morphisms of left modules at degrees [ 0 .. 14 ]>
gap> Assert( 0, IsLeftAcyclic( lehs ) );
gap> Display( lehs );
-------------------------
at homology degree: 14
Z/( 256 )/< |[ 4 ]| >
-------------------------
[ [ 4 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 13
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 12
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 11
Z/( 256 )/< |[ 4 ]| >
-------------------------
[ [ 4 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 10
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 9
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 8
Z/( 256 )/< |[ 4 ]| >
-------------------------
[ [ 4 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 7
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 6
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 5
Z/( 256 )/< |[ 4 ]| >
-------------------------
[ [ 4 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 4
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 3
Z/( 256 )/< |[ 8 ]| >
-------------------------
[ [ 2 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 2
Z/( 256 )/< |[ 4 ]| >
-------------------------
[ [ 8 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 1
Z/( 256 )/< |[ 16 ]| >
-------------------------
[ [ 1 ] ]

modulo [ 256 ]

the map is currently represented by the above 1 x 1 matrix
------------v------------
at homology degree: 0
Z/( 256 )/< |[ 8 ]| >
-------------------------
</pre></div>

<a id="X85CF19B87D1C375F" name="X85CF19B87D1C375F"></a>

<h4>3.2 Commutative Algebra</h4>

<a id="X781B1C0C80529B09" name="X781B1C0C80529B09"></a>

<h5>3.2-1 Eliminate</h5>

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,l,m";
Q[x,y,z,l,m]
gap> var := Indeterminates( R );
[ x, y, z, l, m ]
gap> x := var[1];; y := var[2];; z := var[3];; l := var[4];; m := var[5];;
gap> L := [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ];
[ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ]
gap> e := Eliminate( L, [ l, m ] );
<A non-zero right regular 3 x 1 matrix over an external ring>
gap> Display( e );
4*y+z,
4*x-5*z,
21*z^2-8
gap> I := LeftSubmodule( e );
<A torsion-free (left) ideal given by 3 generators>
gap> Display( I );
4*y+z,
4*x-5*z,
21*z^2-8

A (left) ideal generated by the 3 entries of the above matrix
gap> J := LeftSubmodule( "x+y-z, -2*z-3*y+x, x^2+y^2+z^2-1", R );
<A torsion-free (left) ideal given by 3 generators>
gap> I = J;
true
</pre></div>

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap2.html">[Previous Chapter]</a> <a href="chapBib.html">[Next Chapter]</a> </div>

<div class="chlinkbot">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

<hr />
generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a>
</body>
</html>

quality100%

¤ Dauer der Verarbeitung: 0.30 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.