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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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

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<!-- This is an automatically generated file. -->

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<Chapter Label="Chapter_Examples_and_Tests">

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<Heading>Examples and Tests</Heading>

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<Section Label="Chapter_Examples_and_Tests_Section_Annihilator">

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<Heading>Annihilator</Heading>

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<Example><![CDATA[

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gap> ZZ := HomalgRingOfIntegersInSingular();;

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gap> M1 := AsLeftPresentation( HomalgMatrix( [ [ "2" ] ], ZZ ) );;

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gap> M2 := AsLeftPresentation( HomalgMatrix( [ [ "3" ] ], ZZ ) );;

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gap> M3 := AsLeftPresentation( HomalgMatrix( [ [ "4" ] ], ZZ ) );;

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gap> M := DirectSum( M1, M2, M3 );;

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gap> Display( Annihilator( M ) );

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12

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A monomorphism in Category of left presentations of Z

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gap> M1 := AsRightPresentation( HomalgMatrix( [ [ "2" ] ], ZZ ) );;

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gap> M2 := AsRightPresentation( HomalgMatrix( [ [ "3" ] ], ZZ ) );;

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gap> M3 := AsRightPresentation( HomalgMatrix( [ [ "4" ] ], ZZ ) );;

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gap> M := DirectSum( M1, M2, M3 );;

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gap> Display( Annihilator( M ) );

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12

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A monomorphism in Category of right presentations of Z

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]]></Example>

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</Section>

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<Section Label="Chapter_Examples_and_Tests_Section_Intersection_of_Submodules">

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<Heading>Intersection of Submodules</Heading>

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<Example><![CDATA[

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gap> Q := HomalgFieldOfRationalsInSingular();;

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gap> R := Q * "x,y";

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Q[x,y]

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gap> F := AsLeftPresentation( HomalgMatrix( [ [ 0 ] ], R ) );

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<An object in Category of left presentations of Q[x,y]>

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gap> I1 := AsLeftPresentation( HomalgMatrix( [ [ "x" ] ], R ) );;

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gap> I2 := AsLeftPresentation( HomalgMatrix( [ [ "y" ] ], R ) );;

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gap> Display( I1 );

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x

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An object in Category of left presentations of Q[x,y]

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gap> Display( I2 );

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y

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An object in Category of left presentations of Q[x,y]

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gap> eps1 := PresentationMorphism( F, HomalgMatrix( [ [ 1 ] ], R ), I1 );

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<A morphism in Category of left presentations of Q[x,y]>

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gap> eps2 := PresentationMorphism( F, HomalgMatrix( [ [ 1 ] ], R ), I2 );

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<A morphism in Category of left presentations of Q[x,y]>

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gap> kernelemb1 := KernelEmbedding( eps1 );

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<A monomorphism in Category of left presentations of Q[x,y]>

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gap> kernelemb2 := KernelEmbedding( eps2 );

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<A monomorphism in Category of left presentations of Q[x,y]>

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gap> P := FiberProduct( kernelemb1, kernelemb2 );;

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gap> Display( P );

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(an empty 0 x 1 matrix)

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An object in Category of left presentations of Q[x,y]

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gap> pi1 := ProjectionInFactorOfFiberProduct( [ kernelemb1, kernelemb2 ], 1 );

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<A monomorphism in Category of left presentations of Q[x,y]>

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gap> composite := PreCompose( pi1, kernelemb1 );

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<A monomorphism in Category of left presentations of Q[x,y]>

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gap> Display( composite );

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x*y

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A monomorphism in Category of left presentations of Q[x,y]

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]]></Example>

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</Section>

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<Section Label="Chapter_Examples_and_Tests_Section_Koszul_Complex">

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<Heading>Koszul Complex</Heading>

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<Example><![CDATA[

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gap> Q := HomalgFieldOfRationalsInSingular();;

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gap> R := Q * "x,y,z";;

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gap> M := HomalgMatrix( [ [ "x" ], [ "y" ], [ "z" ] ], 3, 1, R );;

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gap> Ml := AsLeftPresentation( M );;

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gap> eps := CoverByFreeModule( Ml );;

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gap> iota1 := KernelEmbedding( eps );;

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gap> Display( iota1 );

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x,

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y,

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z 

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A monomorphism in Category of left presentations of Q[x,y,z]

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gap> Display( Source( iota1 ) );

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0, -z,y,

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-y,x, 0,

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-z,0, x 

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An object in Category of left presentations of Q[x,y,z]

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gap> pi1 := CoverByFreeModule( Source( iota1 ) );;

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gap> d1 := PreCompose( pi1, iota1 );;

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gap> Display( d1 );

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x,

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y,

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z 

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A morphism in Category of left presentations of Q[x,y,z]

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gap> iota2 := KernelEmbedding( d1 );;

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gap> Display( iota2 );

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0, -z,y,

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-y,x, 0,

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-z,0, x 

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A monomorphism in Category of left presentations of Q[x,y,z]

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gap> Display( Source( iota2 ) );;

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x,z,-y

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An object in Category of left presentations of Q[x,y,z]

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gap> pi2 := CoverByFreeModule( Source( iota2 ) );;

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gap> d2 := PreCompose( pi2, iota2 );;

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gap> Display( d2 );

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0, -z,y,

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-y,x, 0,

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-z,0, x 

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A morphism in Category of left presentations of Q[x,y,z]

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gap> iota3 := KernelEmbedding( d2 );;

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gap> Display( iota3 );

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x,z,-y

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A monomorphism in Category of left presentations of Q[x,y,z]

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gap> Display( Source( iota3 ) );

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(an empty 0 x 1 matrix)

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An object in Category of left presentations of Q[x,y,z]

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gap> pi3 := CoverByFreeModule( Source( iota3 ) );;

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gap> d3 := PreCompose( pi3, iota3 );;

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gap> Display( d3 );

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x,z,-y

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A morphism in Category of left presentations of Q[x,y,z]

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gap> N := HomalgMatrix( [ [ "x" ] ], 1, 1, R );;

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gap> Nl := AsLeftPresentation( N );;

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gap> d2Nl := TensorProductOnMorphisms( d2, IdentityMorphism( Nl ) );;

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gap> d1Nl := TensorProductOnMorphisms( d1, IdentityMorphism( Nl ) );;

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gap> IsZero( PreCompose( d2Nl, d1Nl ) );

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true

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gap> cycles := KernelEmbedding( d1Nl );;

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gap> boundaries := ImageEmbedding( d2Nl );;

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gap> boundaries_in_cyles := LiftAlongMonomorphism( cycles, boundaries );;

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gap> homology := CokernelObject( boundaries_in_cyles );;

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gap> LessGenFunctor := FunctorLessGeneratorsLeft( R );;

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gap> homology := ApplyFunctor( LessGenFunctor, homology );;

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gap> StdBasisFunctor := FunctorStandardModuleLeft( R );;

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gap> homology := ApplyFunctor( StdBasisFunctor, homology );;

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gap> Display( homology );

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z,

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y,

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x 

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An object in Category of left presentations of Q[x,y,z]

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]]></Example>

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</Section>

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<Section Label="Chapter_Examples_and_Tests_Section_Closed_Monoidal_Structure">

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<Heading>Closed Monoidal Structure</Heading>

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<Example><![CDATA[

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gap> R := HomalgRingOfIntegers( );;

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gap> M := AsLeftPresentation( HomalgMatrix( [ [ 2 ] ], 1, 1, R ) );

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<An object in Category of left presentations of Z>

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gap> N := AsLeftPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, R ) );

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<An object in Category of left presentations of Z>

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gap> T := TensorProductOnObjects( M, N );

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<An object in Category of left presentations of Z>

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gap> Display( T );

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[ [  3 ],

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  [  2 ] ]

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An object in Category of left presentations of Z

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gap> IsZero( T );

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true

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gap> H := InternalHomOnObjects( DirectSum( M, M ), DirectSum( M, N ) );

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<An object in Category of left presentations of Z>

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gap> Display( H );

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[ [  -4,  -2 ],

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  [   2,   2 ] ]

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An object in Category of left presentations of Z

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gap> alpha := StandardGeneratorMorphism( H, 1 );

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<A morphism in Category of left presentations of Z>

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gap> l := LambdaElimination( DirectSum( M, M ), DirectSum( M, N ), alpha );

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<A morphism in Category of left presentations of Z>

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gap> IsZero( l );

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false

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gap> Display( l );

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[ [  0,   0 ],

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  [  1,  0 ] ]

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A morphism in Category of left presentations of Z

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gap> alpha2 := StandardGeneratorMorphism( H, 2 );

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<A morphism in Category of left presentations of Z>

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gap> l2 := LambdaElimination( DirectSum( M, M ), DirectSum( M, N ), alpha2 );

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<A morphism in Category of left presentations of Z>

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gap> IsZero( l2 );

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false

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gap> Display( l2 );

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[ [  1,   0 ],

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  [  0,  0 ] ]

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A morphism in Category of left presentations of Z

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]]></Example>

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</Section>

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</Chapter>

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