Quelle chap13.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/modules/doc/chap13.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 (Modules) - Chapter 13: 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="chap13" 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="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chapA.html">A</a> <a href="chapB.html">B</a> <a href="chapC.html">C</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="chap12.html">[Previous Chapter]</a> <a href="chapA.html">[Next Chapter]</a> </div>

<a href="chap13_mj.html">[MathJax on]</a>
<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap13.html#X7A489A5D79DA9E5C">13 Examples</a>
<div class="ContSect"> <a href="chap13.html#X7BB9DE017ECE6E86">13.1 ExtExt</a>

</div>
<div class="ContSect"> <a href="chap13.html#X7EE63228803A04F1">13.2 Purity</a>

</div>
<div class="ContSect"> <a href="chap13.html#X812EF8147AE16E72">13.3 TorExt-Grothendieck</a>

</div>
<div class="ContSect"> <a href="chap13.html#X784BC2567875830B">13.4 TorExt</a>

</div>
</div>

<h3>13 Examples</h3>

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

<h4>13.1 ExtExt</h4>

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

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> N := Hom( zz, M );
<A rank 1 right module on 4 generators satisfying yet unknown relations>
gap> F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, N, "TensorN" );
<The functor TensorN for f.p. modules and their maps over computable rings>
gap> G := LeftDualizingFunctor( zz );;
gap> II_E := GrothendieckSpectralSequence( F, G, M );
<A stable homological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a homological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

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

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

s s
. .

Now the spectral sequence of the bicomplex:

a homological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

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

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

s s
. s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A non-torsion left module presented by 3 relations for 4 generators>
 -1: <A non-zero left module presented by 21 relations for 8 generators>
of
<A non-zero left module presented by 31 relations for 19 generators>>
gap> ByASmallerPresentation( filt );
<An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A rank 1 left module presented by 2 relations for 3 generators>

-1: <A non-zero torsion left module presented by 6 relations for 6 generators>
of
<A rank 1 left module presented by 8 relations for 9 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

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

<h4>13.2 Purity</h4>

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

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> filt := PurityFiltration( M );
<The ascending purity filtration with degrees [ -1 .. 0 ] and graded parts:
 0: <A free left module of rank 1 on a free generator>

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

a homological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

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

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

s .
. .

Now the spectral sequence of the bicomplex:

a homological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

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

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

s .
. s
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
gap> IsIdenticalObj( Range( m ), M );
true
gap> Source( m );
<A non-torsion left module presented by 2 relations for 3 generators (locked)>
gap> Display( last );
[ [ 0, 2, 0 ],
 [ 0, 0, 12 ] ]

Cokernel of the map

Z^(1x2) --> Z^(1x3),

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

Z^(1 x 1)
----------
Degree -1:

Z/< 2 > + Z/< 12 >
</pre></div>

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

<h4>13.3 TorExt-Grothendieck</h4>

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

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, M, "TensorM" );
<The functor TensorM for f.p. modules and their maps over computable rings>
gap> G := LeftDualizingFunctor( zz );;
gap> II_E := GrothendieckSpectralSequence( F, G, M );
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a cohomological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

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

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

s s
. .

Now the spectral sequence of the bicomplex:

a cohomological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

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

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

s s
. s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:

-1: <A non-zero left module presented by yet unknown relations for 6 generator\
s>

0: <A non-zero left module presented by yet unknown relations for 4 generators\
>
of
<A left module presented by yet unknown relations for 14 generators>>
gap> ByASmallerPresentation( filt );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:
 -1: <A non-zero torsion left module presented by 4 relations
 for 4 generators>
 0: <A rank 1 left module presented by 2 relations for 3 generators>
of
<A rank 1 left module presented by 6 relations for 7 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

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

<h4>13.4 TorExt</h4>

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

<div class="example"><pre>
gap> zz := HomalgRingOfIntegers( );
Z
gap> imat := HomalgMatrix( "[ \
> 262, -33, 75, -40, \
> 682, -86, 196, -104, \
> 1186, -151, 341, -180, \
> -1932, 248, -556, 292, \
> 1018, -127, 293, -156 \
> ]", 5, 4, zz );
<A 5 x 4 matrix over an internal ring>
gap> M := LeftPresentation( imat );
<A left module presented by 5 relations for 4 generators>
gap> P := Resolution( M );
<A non-zero right acyclic complex containing a single morphism of left modules\
at degrees [ 0 .. 1 ]>
gap> GP := Hom( P );
<A non-zero acyclic cocomplex containing a single morphism of right modules at\
degrees [ 0 .. 1 ]>
gap> FGP := GP * P;
<A non-zero acyclic cocomplex containing a single morphism of left complexes a\
t degrees [ 0 .. 1 ]>
gap> BC := HomalgBicomplex( FGP );
<A non-zero bicocomplex containing left modules at bidegrees [ 0 .. 1 ]x
[ -1 .. 0 ]>
gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1];
[ 0, 1 ]
gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );
<A stable cohomological spectral sequence with sheets at levels
[ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
[ 0 .. 1 ]>
gap> Display( II_E );
The associated transposed spectral sequence:

a cohomological spectral sequence at bidegrees
[ [ 0 .. 1 ], [ -1 .. 0 ] ]
---------
Level 0:

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

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

s s
. .

Now the spectral sequence of the bicomplex:

a cohomological spectral sequence at bidegrees
[ [ -1 .. 0 ], [ 0 .. 1 ] ]
---------
Level 0:

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

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

s s
. s
gap> filt := FiltrationBySpectralSequence( II_E, 0 );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:

-1: <A non-zero torsion left module presented by yet unknown relations for
 4 generators>
 0: <A rank 1 left module presented by 3 relations for 4 generators>
of
<A left module presented by yet unknown relations for 13 generators>>
gap> ByASmallerPresentation( filt );
<A descending filtration with degrees [ -1 .. 0 ] and graded parts:
 -1: <A non-zero torsion left module presented by 4 relations
 for 4 generators>
 0: <A rank 1 left module presented by 2 relations for 3 generators>
of
<A rank 1 left module presented by 6 relations for 7 generators>>
gap> m := IsomorphismOfFiltration( filt );
<A non-zero isomorphism of left modules>
</pre></div>

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap12.html">[Previous Chapter]</a> <a href="chapA.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="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chapA.html">A</a> <a href="chapB.html">B</a> <a href="chapC.html">C</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

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

Messung
C	H	Z	G
84	100	100	94

¤ Dauer der Verarbeitung: 0.9 Sekunden ¤

Wurzel

Suchen

PVS Prover

Isabelle Prover

Cephes Mathematical Library

Vienna Development Method

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 und die Messung sind noch experimentell.