Goto Chapter: Top 1 2 3 4 5 6 7 8 9 Ind

### 9 Examples and Tests

#### 9.1 Basic Commands

gap> Q := HomalgFieldOfRationals();;
gap> A := VectorSpaceObject( 4, Q );;
gap> B := VectorSpaceObject( 3, Q );;
gap> C := VectorSpaceObject( 2, Q );;
gap> alpha := VectorSpaceMorphism( A,
> HomalgMatrix( [ [ 1, 1, 1 ], [ 0, 1, 1 ],
> [ 1, 0, 1 ], [ 1, 1, 0 ] ], 4, 3, Q ), B );;
gap> gamma := VectorSpaceMorphism( C,
> HomalgMatrix( [ [ -1, 1, -1 ], [ 1, 0, -1 ] ], 2, 3, Q ), B );;
gap> p := ProjectionInFactorOfFiberProduct( [ alpha, gamma ], 1 );;
gap> q := ProjectionInFactorOfFiberProduct( [ alpha, gamma ], 2 );;
gap> PreCompose( AsGeneralizedMorphism( alpha ), GeneralizedInverse( gamma ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> gen1 := PreCompose( AsGeneralizedMorphism( alpha ),
>                        GeneralizedInverse( gamma ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> gen2 := PreCompose( GeneralizedInverse( p ), AsGeneralizedMorphism( q ) );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsCongruentForMorphisms( gen1, gen2 );
true


#### 9.2 Intersection of Nodal Curve and Cusp

We are going to intersect the nodal curve f = y^2 - x^2(x+1) and the cusp g = (x+y)^2 - (y-x)^3. The two curves are arranged in a way such that they intersect at (0,0) with intersection number as high as possible. We are going to compute this intersection number using the definition of the intersection number as the length of the module R/(f,g) localized at (0,0). In order to model modules over the localization of Q[x,y] at (0,0), we use a suitable Serre quotient category. 1 2 1 1 true We are going to intersect the nodal curve f = y^2 - x^2(x+1) and the cusp g = (x+y)^2 - (y-x)^3. The two curves are arranged in a way such that they intersect at (0,0) with intersection number as high as possible. We are going to compute this intersection number using the definition of the intersection number as the length of the module R/(f,g) localized at (0,0). In order to model modules over the localization of Q[x,y] at (0,0), we use a suitable Serre quotient category. 1 2 1 1 true We are going to intersect the nodal curve f = y^2 - x^2(x+1) and the cusp g = (x+y)^2 - (y-x)^3. The two curves are arranged in a way such that they intersect at (0,0) with intersection number as high as possible. We are going to compute this intersection number using the definition of the intersection number as the length of the module R/(f,g) localized at (0,0). In order to model modules over the localization of Q[x,y] at (0,0), we use a suitable Serre quotient category. 1 2 1 1 true

#### 9.3 WrapperCategory

gap> LoadPackage( "LinearAlgebraForCAP" );
true
true
gap> old_generalized_morphism_standard := CAP_INTERNAL!.generalized_morphism_standard;;
gap> SwitchGeneralizedMorphismStandard( "cospan" );
gap> Q := HomalgFieldOfRationals( );
Q
gap> id := HomalgIdentityMatrix( 8, Q );
<An unevaluated 8 x 8 identity matrix over an internal ring>
gap> a := CertainColumns( CertainRows( id, [ 1, 2, 3 ] ), [ 2, 3, 4, 5 ] );
<An unevaluated non-zero 3 x 4 matrix over an internal ring>
gap> b := CertainColumns( CertainRows( id, [ 1, 2, 3, 4 ] ), [ 2, 3, 4, 5, 6 ] );
<An unevaluated non-zero 4 x 5 matrix over an internal ring>
gap> c := CertainColumns( CertainRows( id, [ 1, 2, 3, 4, 5 ] ), [ 3, 4, 5, 6, 7, 8 ] );
<An unevaluated non-zero 5 x 6 matrix over an internal ring>
gap> IsZero( a * b );
false
gap> IsZero( b * c );
false
gap> IsZero( a * b * c );
true
gap> Qmat := MatrixCategory( Q );
Category of matrices over Q
gap> Wrapper := WrapperCategory( Qmat, rec( ) );
WrapperCategory( Category of matrices over Q )
gap> a := a / Wrapper;
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> b := b / Wrapper;
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> c := c / Wrapper;
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> d := CokernelProjection( a );
<An epimorphism in WrapperCategory( Category of matrices over Q )>
gap> e := CokernelColift( a, PreCompose( b, c ) );
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> f := KernelEmbedding( e );
<A monomorphism in WrapperCategory( Category of matrices over Q )>
gap> g := KernelEmbedding( c );
<A monomorphism in WrapperCategory( Category of matrices over Q )>
gap> h := KernelLift( c, PreCompose( a, b ) );
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> i := CokernelProjection( h );
<An epi morphism in WrapperCategory( Category of matrices over Q )>
gap> ff := AsGeneralizedMorphism( f );
<A morphism in Generalized morphism category of
WrapperCategory( Category of matrices over Q ) by cospan>
gap> dd := AsGeneralizedMorphism( d );
<A morphism in Generalized morphism category of
WrapperCategory( Category of matrices over Q ) by cospan>
gap> bb := AsGeneralizedMorphism( b );
<A morphism in Generalized morphism category of
WrapperCategory( Category of matrices over Q ) by cospan>
gap> gg := AsGeneralizedMorphism( g );
<A morphism in Generalized morphism category of
WrapperCategory( Category of matrices over Q ) by cospan>
gap> ii := AsGeneralizedMorphism( i );
<A morphism in Generalized morphism category of
WrapperCategory( Category of matrices over Q ) by cospan>
gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] );
<A morphism in Generalized morphism category of
WrapperCategory( Category of matrices over Q ) by cospan>
gap> s := HonestRepresentative( ss );
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> j := KernelObjectFunctorial( b, d, e );
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> k := CokernelObjectFunctorial( h, g, b );
<A morphism in WrapperCategory( Category of matrices over Q )>
gap> HK := HomologyObject( j, s );
<An object in WrapperCategory( Category of matrices over Q )>
gap> HC := HomologyObject( s, k );
<An object in WrapperCategory( Category of matrices over Q )>
gap> SwitchGeneralizedMorphismStandard( old_generalized_morphism_standard );


#### 9.4 Sweep

\href{https://terrytao.wordpress.com/2015/10/07/sweeping-a-matrix-rotates-its-graph/}{\textrm{Geometric interpretation of sweeping a matrix by Terence Tao.}}

gap> Q := HomalgFieldOfRationals();;
gap> V := VectorSpaceObject( 3, Q );;
gap> mat := HomalgMatrix( [ [ 9, 8, 7 ], [ 6, 5, 4 ], [ 3, 2, 1 ] ], 3, 3, Q );;
gap> alpha := VectorSpaceMorphism( V, mat, V );;
gap> graph := FiberProductEmbeddingInDirectSum(
>             [ alpha, IdentityMorphism( V ) ] );;
gap> Display( graph );
[ [     1,    -2,     1,     0,     0,     0 ],
[  -4/3,   7/3,     0,     2,     1,     0 ],
[   5/3,  -8/3,     0,    -1,     0,     1 ] ]

A morphism in Category of matrices over Q
gap> D := DirectSum( V, V );;
gap> rotmat := HomalgMatrix( [ [ 0, 0, 0, -1, 0, 0 ],
>                              [ 0, 1, 0, 0, 0, 0 ],
>                              [ 0, 0, 1, 0, 0, 0 ],
>                              [ 1, 0, 0, 0, 0, 0 ],
>                              [ 0, 0, 0, 0, 1, 0 ],
>                              [ 0, 0, 0, 0, 0, 1 ] ],
>                              6, 6, Q );;
gap> rot := VectorSpaceMorphism( D, rotmat, D );;
gap> p := PreCompose( graph, rot );;
gap> Display( p );
[ [     0,    -2,     1,    -1,     0,     0 ],
[     2,   7/3,     0,   4/3,     1,     0 ],
[    -1,  -8/3,     0,  -5/3,     0,     1 ] ]

A morphism in Category of matrices over Q
gap> pi1 := ProjectionInFactorOfDirectSum( [ V, V ], 1 );;
gap> pi2 := ProjectionInFactorOfDirectSum( [ V, V ], 2 );;
gap> reversed_arrow := PreCompose( p, pi1 );;
gap> arrow := PreCompose( p, pi2 );;
gap> g := GeneralizedMorphismBySpan( reversed_arrow, arrow );;
gap> IsHonest( g );
true
gap> sweep_1_alpha := HonestRepresentative( g );;
gap> Display( sweep_1_alpha );
[ [  -1/9,   8/9,   7/9 ],
[   2/3,  -1/3,  -2/3 ],
[   1/3,  -2/3,  -4/3 ] ]

A morphism in Category of matrices over Q
gap> Display( alpha );
[ [  9,  8,  7 ],
[  6,  5,  4 ],
[  3,  2,  1 ] ]

A morphism in Category of matrices over Q


#### 9.5 Generalized Morphisms Category

gap> Q := HomalgFieldOfRationals();
Q
gap> B := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> C := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> B_1 := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> C_1 := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> c1_source_aid := VectorSpaceMorphism( B_1, [ [ 1, 0 ] ], B );
<A morphism in Category of matrices over Q>
gap> SetIsSubobject( c1_source_aid, true );
gap> c1_range_aid := VectorSpaceMorphism( C, [ [ 1, 0 ], [ 0, 1 ], [ 0, 0 ] ], C_1 );
<A morphism in Category of matrices over Q>
gap> SetIsFactorobject( c1_range_aid, true );
gap> c1_associated := VectorSpaceMorphism( B_1, [ [ 1, 1 ] ], C_1 );
<A morphism in Category of matrices over Q>
gap> c1 := GeneralizedMorphism( c1_source_aid, c1_associated, c1_range_aid );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> B_2 := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> C_2 := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> c2_source_aid := VectorSpaceMorphism( B_2, [ [ 2, 0 ] ], B );
<A morphism in Category of matrices over Q>
gap> SetIsSubobject( c2_source_aid, true );
gap> c2_range_aid := VectorSpaceMorphism( C, [ [ 3, 0 ], [ 0, 3 ], [ 0, 0 ] ], C_2 );
<A morphism in Category of matrices over Q>
gap> SetIsFactorobject( c2_range_aid, true );
gap> c2_associated := VectorSpaceMorphism( B_2, [ [ 6, 6 ] ], C_2 );
<A morphism in Category of matrices over Q>
gap> c2 := GeneralizedMorphism( c2_source_aid, c2_associated, c2_range_aid );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsCongruentForMorphisms( c1, c2 );
true
gap> IsCongruentForMorphisms( c1, c1 );
true
gap> c3_associated := VectorSpaceMorphism( B_1, [ [ 2, 2 ] ], C_1 );
<A morphism in Category of matrices over Q>
gap> c3 := GeneralizedMorphism( c1_source_aid, c3_associated, c1_range_aid );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsCongruentForMorphisms( c1, c3 );
false
gap> IsCongruentForMorphisms( c2, c3 );
false
gap> c1 + c2;
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> Arrow( c1 + c2 );
<A morphism in Category of matrices over Q>


First composition test:

gap> Q := HomalgFieldOfRationals();
Q
gap> A := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> B := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> C := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> phi_tilde_associated := VectorSpaceMorphism( A, [ [ 1, 2, 0 ] ], C );
<A morphism in Category of matrices over Q>
gap> phi_tilde_source_aid := VectorSpaceMorphism( A, [ [ 1, 2 ] ], B );
<A morphism in Category of matrices over Q>
gap> phi_tilde := GeneralizedMorphismWithSourceAid( phi_tilde_source_aid, phi_tilde_associated );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> psi_tilde_associated := IdentityMorphism( B );
<An identity morphism in Category of matrices over Q>
gap> psi_tilde_source_aid := VectorSpaceMorphism( B, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], C );
<A morphism in Category of matrices over Q>
gap> psi_tilde := GeneralizedMorphismWithSourceAid( psi_tilde_source_aid, psi_tilde_associated );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> composition := PreCompose( phi_tilde, psi_tilde );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> Arrow( composition );
<A morphism in Category of matrices over Q>
gap> SourceAid( composition );
<A morphism in Category of matrices over Q>
gap> RangeAid( composition );
<An identity morphism in Category of matrices over Q>


Second composition test

gap> Q := HomalgFieldOfRationals();
Q
gap> A := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> B := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> C := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> phi2_tilde_associated := VectorSpaceMorphism( A, [ [ 1, 5 ] ], B );
<A morphism in Category of matrices over Q>
gap> phi2_tilde_range_aid := VectorSpaceMorphism( C, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ], B );
<A morphism in Category of matrices over Q>
gap> phi2_tilde := GeneralizedMorphismWithRangeAid( phi2_tilde_associated, phi2_tilde_range_aid );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> psi2_tilde_associated := VectorSpaceMorphism( C, [ [ 1 ], [ 3 ], [ 4 ] ], A );
<A morphism in Category of matrices over Q>
gap> psi2_tilde_range_aid := VectorSpaceMorphism( B, [ [ 1 ], [ 1 ] ], A );
<A morphism in Category of matrices over Q>
gap> psi2_tilde := GeneralizedMorphismWithRangeAid( psi2_tilde_associated, psi2_tilde_range_aid );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> composition2 := PreCompose( phi2_tilde, psi2_tilde );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> Arrow( composition2 );
<A morphism in Category of matrices over Q>
gap> RangeAid( composition2 );
<A morphism in Category of matrices over Q>
gap> SourceAid( composition2 );
<An identity morphism in Category of matrices over Q>


Third composition test

gap> Q := HomalgFieldOfRationals();
Q
gap> A := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> Asub := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> B := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> Bfac := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> Bsub := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> C := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> Cfac := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> Asub_into_A := VectorSpaceMorphism( Asub, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], A );
<A morphism in Category of matrices over Q>
gap> Asub_to_Bfac := VectorSpaceMorphism( Asub, [ [ 1 ], [ 1 ] ], Bfac );
<A morphism in Category of matrices over Q>
gap> B_onto_Bfac := VectorSpaceMorphism( B, [ [ 1 ], [ 1 ], [ 1 ] ], Bfac );
<A morphism in Category of matrices over Q>
gap> Bsub_into_B := VectorSpaceMorphism( Bsub, [ [ 2, 2, 0 ], [ 0, 2, 2 ] ], B );
<A morphism in Category of matrices over Q>
gap> Bsub_to_Cfac := VectorSpaceMorphism( Bsub, [ [ 3 ], [ 0 ] ], Cfac );
<A morphism in Category of matrices over Q>
gap> C_onto_Cfac := VectorSpaceMorphism( C, [ [ 1 ], [ 2 ], [ 3 ] ], Cfac );
<A morphism in Category of matrices over Q>
gap> generalized_morphism1 := GeneralizedMorphism( Asub_into_A, Asub_to_Bfac, B_onto_Bfac );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> generalized_morphism2 := GeneralizedMorphism( Bsub_into_B, Bsub_to_Cfac, C_onto_Cfac );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsWellDefined( generalized_morphism1 );
true
gap> IsWellDefined( generalized_morphism2 );
true
gap> p := PreCompose( generalized_morphism1, generalized_morphism2 );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> SourceAid( p );
<A morphism in Category of matrices over Q>
gap> Arrow( p );
<A morphism in Category of matrices over Q>
gap> RangeAid( p );
<A morphism in Category of matrices over Q>
gap> A := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> Asub := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> B := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> Bfac := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> Bsub := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> C := VectorSpaceObject( 3, Q );
<A vector space object over Q of dimension 3>
gap> Cfac := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> Bsub_to_Cfac := VectorSpaceMorphism( Bsub, [ [ 3, 3 ], [ 0, 0 ] ], Cfac );
<A morphism in Category of matrices over Q>
gap> C_onto_Cfac := VectorSpaceMorphism( C, [ [ 1, 0 ], [ 0, 2 ], [ 3, 3 ] ], Cfac );
<A morphism in Category of matrices over Q>
gap> generalized_morphism1 := GeneralizedMorphism( Asub_into_A, Asub_to_Bfac, B_onto_Bfac );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> generalized_morphism2 := GeneralizedMorphism( Bsub_into_B, Bsub_to_Cfac, C_onto_Cfac );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsWellDefined( generalized_morphism1 );
true
gap> IsWellDefined( generalized_morphism2 );
true
gap> p := PreCompose( generalized_morphism1, generalized_morphism2 );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> SourceAid( p );
<A morphism in Category of matrices over Q>
gap> Arrow( p );
<A morphism in Category of matrices over Q>
gap> RangeAid( p );
<A morphism in Category of matrices over Q>


Honest representative test

gap> Q := HomalgFieldOfRationals();
Q
gap> A := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> B := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> phi_tilde_source_aid := VectorSpaceMorphism( A, [ [ 2 ] ], A );
<A morphism in Category of matrices over Q>
gap> phi_tilde_associated := VectorSpaceMorphism( A, [ [ 1, 1 ] ], B );
<A morphism in Category of matrices over Q>
gap> phi_tilde_range_aid := VectorSpaceMorphism( B, [ [ 1, 2 ], [ 3, 4 ] ], B );
<A morphism in Category of matrices over Q>
gap> phi_tilde := GeneralizedMorphism( phi_tilde_source_aid, phi_tilde_associated, phi_tilde_range_aid );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> HonestRepresentative( phi_tilde );
<A morphism in Category of matrices over Q>
gap> IsWellDefined( phi_tilde );
true
gap> IsWellDefined( psi_tilde );
true


#### 9.6 IsWellDefined

gap> Q := HomalgFieldOfRationals();
Q
gap> A := VectorSpaceObject( 1, Q );
<A vector space object over Q of dimension 1>
gap> B := VectorSpaceObject( 2, Q );
<A vector space object over Q of dimension 2>
gap> alpha := VectorSpaceMorphism( A, [ [ 1, 2 ] ], B );
<A morphism in Category of matrices over Q>
gap> g := GeneralizedMorphism( alpha, alpha, alpha );
<A morphism in Generalized morphism category of Category of matrices over Q>
gap> IsWellDefined( alpha );
true
gap> IsWellDefined( g );
true
gap> IsEqualForObjects( A, B );
false

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