Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/Sources/formale Sprachen/GAP/pkg/gradedmodules/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 22.11.2024 mit Größe 23 kB

SSL chap5.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/gradedmodules/doc/chap5.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 (GradedModules) - Chapter 5: 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="chap5" 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="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="chap4.html">[Previous Chapter]</a> <a href="chapBib.html">[Next Chapter]</a> </div>

<a href="chap5_mj.html">[MathJax on]</a>
<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chap5.html#X7A489A5D79DA9E5C">5 Examples</a>
<div class="ContSect"> <a href="chap5.html#X81A7E0D380CE7F31">5.1 Betti Diagrams</a>

<div class="ContSSBlock">
 <a href="chap5.html#X8441906E83F6845D">5.1-1 DE-2.2</a>

 <a href="chap5.html#X7E32106D7B13B8D9">5.1-2 DE-Code</a>

 <a href="chap5.html#X793A69C4805C6819">5.1-3 Schenck-3.2</a>

 <a href="chap5.html#X7E8F44338461DC08">5.1-4 Schenck-8.3</a>

 <a href="chap5.html#X7B672C498385F92F">5.1-5 Schenck-8.3.3</a>

</div></div>
<div class="ContSect"> <a href="chap5.html#X85CF19B87D1C375F">5.2 Commutative Algebra</a>

<div class="ContSSBlock">
 <a href="chap5.html#X7EA4CC697C01E080">5.2-1 Saturate</a>

</div></div>
<div class="ContSect"> <a href="chap5.html#X86AF934C83004BF2">5.3 Global Section Modules of the Induced Sheaves</a>

<div class="ContSSBlock">
 <a href="chap5.html#X87EE931187E2226C">5.3-1 Examples of the ModuleOfGlobalSections Functor and Purity Filtrations</a>

 <a href="chap5.html#X7DD8F76D7A4206E3">5.3-2 Horrocks Mumford bundle</a>

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

<h3>5 Examples</h3>

<a id="X81A7E0D380CE7F31" name="X81A7E0D380CE7F31"></a>

<h4>5.1 Betti Diagrams</h4>

<a id="X8441906E83F6845D" name="X8441906E83F6845D"></a>

<h5>5.1-1 DE-2.2</h5>

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0,x1,x2";;
gap> S := GradedRing( R );;
gap> mat := HomalgMatrix( "[ x0^2, x1^2, x2^2 ]", 1, 3, S ); 
<A 1 x 3 matrix over a graded ring>
gap> M := RightPresentationWithDegrees( mat, S );
<A graded cyclic right module on a cyclic generator satisfying 3 relations>
gap> M := RightPresentationWithDegrees( mat );
<A graded cyclic right module on a cyclic generator satisfying 3 relations>
gap> d := Resolution( M );
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
gap> betti := BettiTable( d );
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
gap> Display( betti );
total: 1 3 3 1
----------------
 0: 1 . . .
 1: . 3 . .
 2: . . 3 .
 3: . . . 1
----------------
degree: 0 1 2 3
gap> ## we are still below the Castelnuovo-Mumford regularity, which is 3:
gap> M2 := SubmoduleGeneratedByHomogeneousPart( 2, M );
<A graded torsion right submodule given by 3 generators>
gap> d2 := Resolution( M2 );
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
gap> betti2 := BettiTable( d2 );
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
gap> Display( betti2 );
total: 3 8 6 1
----------------
 2: 3 8 6 .
 3: . . . 1
----------------
degree: 0 1 2 3
</pre></div>

<a id="X7E32106D7B13B8D9" name="X7E32106D7B13B8D9"></a>

<h5>5.1-2 DE-Code</h5>

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0,x1,x2";;
gap> S := GradedRing( R );;
gap> mat := HomalgMatrix( "[ x0^2, x1^2 ]", 1, 2, S );
<A 1 x 2 matrix over a graded ring>
gap> M := RightPresentationWithDegrees( mat, S );
<A graded cyclic right module on a cyclic generator satisfying 2 relations>
gap> d := Resolution( M );
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
gap> betti := BettiTable( d );
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
gap> Display( betti );
total: 1 2 1
--------------
 0: 1 . .
 1: . 2 .
 2: . . 1
--------------
degree: 0 1 2
gap> m := SubmoduleGeneratedByHomogeneousPart( 2, M );
<A graded torsion right submodule given by 4 generators>
gap> d2 := Resolution( m );
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
gap> betti2 := BettiTable( d2 );
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
gap> Display( betti2 );
 2: 4 8 4
--------------
degree: 0 1 2
</pre></div>

<a id="X793A69C4805C6819" name="X793A69C4805C6819"></a>

<h5>5.1-3 Schenck-3.2</h5>

This is an example from Section 3.2 in <a href="chapBib.html#biBSch">[Sch03]</a>.

<div class="example"><pre>
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> mmat := HomalgMatrix( "[ x, x^3 + y^3 + z^3 ]", 1, 2, Qxyz );
<A 1 x 2 matrix over an external ring>
gap> S := GradedRing( Qxyz );;
gap> M := RightPresentationWithDegrees( mmat, S );
<A graded cyclic right module on a cyclic generator satisfying 2 relations>
gap> Mr := Resolution( M );
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
gap> bettiM := BettiTable( Mr );
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
gap> Display( bettiM );
total: 1 2 1
--------------
 0: 1 1 .
 1: . . .
 2: . 1 1
--------------
degree: 0 1 2
gap> R := GradedRing( CoefficientsRing( S ) * "x,y,z,w" );;
gap> nmat := HomalgMatrix( "[ z^2 - y*w, y*z - x*w, y^2 - x*z ]", 1, 3, R );
<A 1 x 3 matrix over a graded ring>
gap> N := RightPresentationWithDegrees( nmat );
<A graded cyclic right module on a cyclic generator satisfying 3 relations>
gap> Nr := Resolution( N );
<A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>
gap> bettiN := BettiTable( Nr );
<A Betti diagram of <A right acyclic complex containing
2 morphisms of graded right modules at degrees [ 0 .. 2 ]>>
gap> Display( bettiN );
total: 1 3 2
--------------
 0: 1 . .
 1: . 3 2
--------------
degree: 0 1 2
</pre></div>

<a id="X7E8F44338461DC08" name="X7E8F44338461DC08"></a>

<h5>5.1-4 Schenck-8.3</h5>

This is an example from Section 8.3 in <a href="chapBib.html#biBSch">[Sch03]</a>.

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,w";;
gap> S := GradedRing( R );;
gap> jmat := HomalgMatrix( "[ z*w, x*w, y*z, x*y, x^3*z - x*z^3 ]", 1, 5, S );
<A 1 x 5 matrix over a graded ring>
gap> J := RightPresentationWithDegrees( jmat );
<A graded cyclic right module on a cyclic generator satisfying 5 relations>
gap> Jr := Resolution( J );
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
gap> betti := BettiTable( Jr );
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
gap> Display( betti );
total: 1 5 6 2
----------------
 0: 1 . . .
 1: . 4 4 1
 2: . . . .
 3: . 1 2 1
----------------
degree: 0 1 2 3
</pre></div>

<a id="X7B672C498385F92F" name="X7B672C498385F92F"></a>

<h5>5.1-5 Schenck-8.3.3</h5>

This is Exercise 8.3.3 in <a href="chapBib.html#biBSch">[Sch03]</a>.

<div class="example"><pre>
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( Qxyz );;
gap> mat := HomalgMatrix( "[ x*y*z, x*y^2, x^2*z, x^2*y, x^3 ]", 1, 5, S );
<A 1 x 5 matrix over a graded ring>
gap> M := RightPresentationWithDegrees( mat, S );
<A graded cyclic right module on a cyclic generator satisfying 5 relations>
gap> Mr := Resolution( M );
<A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>
gap> betti := BettiTable( Mr );
<A Betti diagram of <A right acyclic complex containing
3 morphisms of graded right modules at degrees [ 0 .. 3 ]>>
gap> Display( betti );
total: 1 5 6 2
----------------
 0: 1 . . .
 1: . . . .
 2: . 5 6 2
----------------
degree: 0 1 2 3
</pre></div>

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

<h4>5.2 Commutative Algebra</h4>

<a id="X7EA4CC697C01E080" name="X7EA4CC697C01E080"></a>

<h5>5.2-1 Saturate</h5>

<div class="example"><pre>
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;
gap> S := GradedRing( R );;
gap> m := GradedLeftSubmodule( "x,y,z", S );
<A graded torsion-free (left) ideal given by 3 generators>
gap> I := Intersect( m^3, GradedLeftSubmodule( "x", S ) );
<A graded torsion-free (left) ideal given by 6 generators>
gap> NrRelations( I );
8
gap> Im := SubobjectQuotient( I, m );
<A graded torsion-free rank 1 (left) ideal given by 3 generators>
gap> I_m := Saturate( I, m );
<A graded principal (left) ideal of rank 1 on a free generator>
gap> Is := Saturate( I );
<A graded principal (left) ideal of rank 1 on a free generator>
gap> Assert( 0, Is = I_m );
</pre></div>

<a id="X86AF934C83004BF2" name="X86AF934C83004BF2"></a>

<h4>5.3 Global Section Modules of the Induced Sheaves</h4>

<a id="X87EE931187E2226C" name="X87EE931187E2226C"></a>

<h5>5.3-1 Examples of the ModuleOfGlobalSections Functor and Purity Filtrations</h5>

<div class="example"><pre>
gap> LoadPackage( "GradedRingForHomalg" );;
gap> Qxyzt := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,t";;
gap> S := GradedRing( Qxyzt );;
gap> 
gap> wmat := HomalgMatrix( "[ \
> x*y, y*z, z*t, 0, 0, 0,\
> x^3*z,x^2*z^2,0, x*z^2*t, -z^2*t^2, 0,\
> x^4, x^3*z, 0, x^2*z*t, -x*z*t^2, 0,\
> 0, 0, x*y, -y^2, x^2-t^2, 0,\
> 0, 0, x^2*z, -x*y*z, y*z*t, 0,\
> 0, 0, x^2*y-x^2*t,-x*y^2+x*y*t,y^2*t-y*t^2,0,\
> 0, 0, 0, 0, -1, 1 \
> ]", 7, 6, Qxyzt );;
gap> 
gap> LoadPackage( "GradedModules" );;
gap> wmor := GradedMap( wmat, "free", "free", "left", S );;
gap> IsMorphism( wmor );;
gap> W := LeftPresentationWithDegrees( wmat, S );;
gap> HW := ModuleOfGlobalSections( W );
<A graded left module presented by yet unknown relations for 6 generators>
gap> LinearStrandOfTateResolution( W, 0,4 );
<A cocomplex containing 4 morphisms of graded left modules at degrees
[ 0 .. 4 ]>
gap> purity_iso := IsomorphismOfFiltration( PurityFiltration( W ) );
<A non-zero isomorphism of graded left modules>
gap> Hpurity_iso := ModuleOfGlobalSections( purity_iso );
<An isomorphism of graded left modules>
gap> ModuleOfGlobalSections( wmor );
<A homomorphism of graded left modules>
gap> NaturalMapToModuleOfGlobalSections( W );
<A homomorphism of graded left modules>
</pre></div>

<a id="X7DD8F76D7A4206E3" name="X7DD8F76D7A4206E3"></a>

<h5>5.3-2 Horrocks Mumford bundle</h5>

This example computes the global sections module of the Horrocks-Mumford bundle.

<div class="example"><pre>
gap> LoadPackage( "GradedRingForHomalg" );;
gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0..x4";;
gap> S := GradedRing( R );;
gap> A := KoszulDualRing( S, "e0..e4" );;
gap> LoadPackage( "GradedModules" );;
gap> mat := HomalgMatrix( "[ \
> e1*e4, e2*e0, e3*e1, e4*e2, e0*e3, \
> e2*e3, e3*e4, e4*e0, e0*e1, e1*e2 \
> ]",
> 2, 5, A );
<A 2 x 5 matrix over a graded ring>
gap> phi := GradedMap( mat, "free", "free", "left", A );;
gap> IsMorphism( phi );
true
gap> M := GuessModuleOfGlobalSectionsFromATateMap( 2, phi );
#I GuessModuleOfGlobalSectionsFromATateMap uses a heuristic for efficiency;
please check the correctness of the following result

<A graded left module presented by yet unknown relations for 19 generators>
gap> IsPure( M );
true
gap> Rank( M );
2
gap> Display( BettiTable( Resolution( M ) ) );
total: 19 35 20 2
--------------------
 3: 4 . . .
 4: 15 35 20 .
 5: . . . 2
--------------------
degree: 0 1 2 3
gap> Display( BettiTable( TateResolution( M, -5, 5 ) ) );
total: 100 37 14 10 5 2 5 10 14 37 100 ? ? ? ?
----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
 4: 100 35 4 . . . . . . . . 0 0 0 0
 3: * . 2 10 10 5 . . . . . . 0 0 0
 2: * * . . . . . 2 . . . . . 0 0
 1: * * * . . . . . . 5 10 10 2 . 0
 0: * * * * . . . . . . . . 4 35 100
----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---S
twist: -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-------------------------------------------------------------------
Euler: 100 35 2 -10 -10 -5 0 2 0 -5 -10 -10 2 35 100
gap> M;
<A graded reflexive non-projective rank 2 left module presented by 99 \
relations for 19 generators>
gap> P := ElementOfGrothendieckGroup( M );
( 2*O_{P^4} - 1*O_{P^3} - 4*O_{P^2} - 2*O_{P^1} ) -> P^4
gap> P!.DisplayTwistedCoefficients := true;
true
gap> P;
( 2*O(-3) - 10*O(-2) + 15*O(-1) - 5*O(0) ) -> P^4
gap> chi := HilbertPolynomial( M );
1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5
gap> c := ChernPolynomial( M );
( 2 | 1-h+4*h^2 ) -> P^4
gap> ChernPolynomial( M * S^3 );
( 2 | 1+5*h+10*h^2 ) -> P^4
gap> ch := ChernCharacter( M );
[ 2-u-7*u^2/2!+11*u^3/3!+17*u^4/4! ] -> P^4
gap> HilbertPolynomial( ch );
1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5
gap> List( [ -8 .. 7 ], i -> Value( chi, i ) );
[ 35, 2, -10, -10, -5, 0, 2, 0, -5, -10, -10, 2, 35, 100, 210, 380 ]
gap> HF := HilbertFunction( M );
function( t ) ... end
gap> List( [ 0 .. 7 ], HF );
[ 0, 0, 0, 4, 35, 100, 210, 380 ]
gap> IndexOfRegularity( M );
4
gap> DataOfHilbertFunction( M );
[ [ [ 4 ], [ 3 ] ], 1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5 ]
</pre></div>

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap4.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="chap4.html">4</a> <a href="chap5.html">5</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>

quality100%

¤ Dauer der Verarbeitung: 0.17 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.