gap> Q := HomalgFieldOfRationals();; gap> vec := MatrixCategory( Q );; gap> A := MatrixCategoryObject( vec, 4 );; gap> B := MatrixCategoryObject( vec, 3 );; gap> C := MatrixCategoryObject( vec, 2 );; 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 gap> # @drop_example_in_Julia (was moved manually)
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
gap> LoadPackage( "LinearAlgebraForCAP" ); true gap> LoadPackage( "GeneralizedMorphismsForCAP", false ); true gap> old_generalized_morphism_standard := CAP_INTERNAL_FIND_CORRECT_GENERALIZED_CATEGORY_TYPE( CAP_INTERNAL );; 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 ); gap> # @drop_example_in_Julia (was moved manually)
\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> vec := MatrixCategory( Q );; gap> V := MatrixCategoryObject( vec, 3 );; 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 gap> # @drop_example_in_Julia (was moved manually)
gap> Q := HomalgFieldOfRationals(); Q gap> vec := MatrixCategory( Q );; gap> B := MatrixCategoryObject( vec, 2 ); <A vector space object over Q of dimension 2> gap> C := MatrixCategoryObject( vec, 3 ); <A vector space object over Q of dimension 3> gap> B_1 := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> C_1 := MatrixCategoryObject( vec, 2 ); <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 := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> C_2 := MatrixCategoryObject( vec, 2 ); <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> gap> # @drop_example_in_Julia (was moved manually)
First composition test:
gap> Q := HomalgFieldOfRationals(); Q gap> vec := MatrixCategory( Q );; gap> A := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> B := MatrixCategoryObject( vec, 2 ); <A vector space object over Q of dimension 2> gap> C := MatrixCategoryObject( vec, 3 ); <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> gap> # @drop_example_in_Julia (was moved manually)
Second composition test
gap> Q := HomalgFieldOfRationals(); Q gap> vec := MatrixCategory( Q );; gap> A := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> B := MatrixCategoryObject( vec, 2 ); <A vector space object over Q of dimension 2> gap> C := MatrixCategoryObject( vec, 3 ); <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> gap> # @drop_example_in_Julia (was moved manually)
Third composition test
gap> Q := HomalgFieldOfRationals(); Q gap> vec := MatrixCategory( Q );; gap> A := MatrixCategoryObject( vec, 3 ); <A vector space object over Q of dimension 3> gap> Asub := MatrixCategoryObject( vec, 2 ); <A vector space object over Q of dimension 2> gap> B := MatrixCategoryObject( vec, 3 ); <A vector space object over Q of dimension 3> gap> Bfac := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> Bsub := MatrixCategoryObject( vec, 2 ); <A vector space object over Q of dimension 2> gap> C := MatrixCategoryObject( vec, 3 ); <A vector space object over Q of dimension 3> gap> Cfac := MatrixCategoryObject( vec, 1 ); <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 := MatrixCategoryObject( vec, 3 ); <A vector space object over Q of dimension 3> gap> Asub := MatrixCategoryObject( vec, 2 ); <A vector space object over Q of dimension 2> gap> B := MatrixCategoryObject( vec, 3 ); <A vector space object over Q of dimension 3> gap> Bfac := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> Bsub := MatrixCategoryObject( vec, 2 ); <A vector space object over Q of dimension 2> gap> C := MatrixCategoryObject( vec, 3 ); <A vector space object over Q of dimension 3> gap> Cfac := MatrixCategoryObject( vec, 2 ); <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> gap> # @drop_example_in_Julia (was moved manually)
Honest representative test
gap> Q := HomalgFieldOfRationals(); Q gap> vec := MatrixCategory( Q );; gap> A := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> B := MatrixCategoryObject( vec, 2 ); <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 gap> # @drop_example_in_Julia (was moved manually)
gap> Q := HomalgFieldOfRationals(); Q gap> vec := MatrixCategory( Q );; gap> A := MatrixCategoryObject( vec, 1 ); <A vector space object over Q of dimension 1> gap> B := MatrixCategoryObject( vec, 2 ); <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 gap> # @drop_example_in_Julia (was moved manually)
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