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<div class="ChapSects"><a href="chap5_mj.html#X7A489A5D79DA9E5C">5 Examples</a>
<div class="ContSect"> <a href="chap5_mj.html#X81A7E0D380CE7F31">5.1 Betti Diagrams</a>

<div class="ContSSBlock">
 <a href="chap5_mj.html#X8441906E83F6845D">5.1-1 DE-2.2</a>

 <a href="chap5_mj.html#X7E32106D7B13B8D9">5.1-2 DE-Code</a>

 <a href="chap5_mj.html#X793A69C4805C6819">5.1-3 Schenck-3.2</a>

 <a href="chap5_mj.html#X7E8F44338461DC08">5.1-4 Schenck-8.3</a>

 <a href="chap5_mj.html#X7B672C498385F92F">5.1-5 Schenck-8.3.3</a>

</div></div>
<div class="ContSect"> <a href="chap5_mj.html#X85CF19B87D1C375F">5.2 Commutative Algebra</a>

<div class="ContSSBlock">
 <a href="chap5_mj.html#X7EA4CC697C01E080">5.2-1 Saturate</a>

</div></div>
<div class="ContSect"> <a href="chap5_mj.html#X86AF934C83004BF2">5.3 Global Section Modules of the Induced Sheaves</a>

<div class="ContSSBlock">
 <a href="chap5_mj.html#X87EE931187E2226C">5.3-1 Examples of the ModuleOfGlobalSections Functor and Purity Filtrations</a>

 <a href="chap5_mj.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_mj.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_mj.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_mj.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>

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