gap> S := HomalgRingOfIntegers(); Z gap> rows := CategoryOfRows( S ); Rows( Z ) gap> obj1 := CategoryOfRowsObject( 2, rows ); <A row module over Z of rank 2> gap> obj2 := CategoryOfRowsObject( 8, rows ); <A row module over Z of rank 8>
gap> obj3 := CategoryOfRowsObject( 1, rows ); <A row module over Z of rank 1> gap> IsWellDefined( obj1 ); true gap> obj4 := CategoryOfRowsObject( 2, rows ); <A row module over Z of rank 2> gap> mor := CategoryOfRowsMorphism( obj3, HomalgMatrix( [[1,2]], S ), obj4 ); <A morphism in Rows( Z )> gap> IsWellDefined( mor ); true
gap> Display( Source( mor ) ); A row module over Z of rank 1 gap> Display( Range( mor ) ); A row module over Z of rank 2 gap> Display( UnderlyingMatrix( mor ) ); [ [ 1, 2 ] ]
gap> Size( AdditiveGenerators( rows ) ); 1 gap> ZeroObject( rows ); <A row module over Z of rank 0> gap> obj5 := CategoryOfRowsObject( 2, rows ); <A row module over Z of rank 2>
gap> Display( ZeroMorphism( ZeroObject( rows ), obj5 ) ); A zero, split monomorphism in Rows( Z ) Source: A row module over Z of rank 0 Matrix: (an empty 0 x 2 matrix) Range: A row module over Z of rank 2
gap> obj6 := CategoryOfRowsObject( 1, rows ); <A row module over Z of rank 1>
gap> Display( IdentityMorphism( obj6 ) ); An identity morphism in Rows( Z ) Source: A row module over Z of rank 1 Matrix: [ [ 1 ] ] Range: A row module over Z of rank 1
gap> directSum := DirectSum( [ obj5, obj6 ] ); <A row module over Z of rank 3>
gap> Display( directSum ); A row module over Z of rank 3
gap> i1 := InjectionOfCofactorOfDirectSum( [ obj5, obj6 ], 1 ); <A morphism in Rows( Z )>
gap> Display( i1 ); A morphism in Rows( Z ) Source: A row module over Z of rank 2 Matrix: [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] Range: A row module over Z of rank 3
gap> i2 := InjectionOfCofactorOfDirectSum( [ obj5, obj6 ], 2 ); <A morphism in Rows( Z )>
gap> Display( i2 ); A morphism in Rows( Z ) Source: A row module over Z of rank 1 Matrix: [ [ 0, 0, 1 ] ] Range: A row module over Z of rank 3
gap> proj1 := ProjectionInFactorOfDirectSum( [ obj5, obj6 ], 1 ); <A morphism in Rows( Z )>
gap> Display( proj1 ); A morphism in Rows( Z ) Source: A row module over Z of rank 3 Matrix: [ [ 1, 0 ], [ 0, 1 ], [ 0, 0 ] ] Range: A row module over Z of rank 2
gap> proj2 := ProjectionInFactorOfDirectSum( [ obj5, obj6 ], 2 ); <A morphism in Rows( Z )>
gap> Display( proj2 ); A morphism in Rows( Z ) Source: A row module over Z of rank 3 Matrix: [ [ 0 ], [ 0 ], [ 1 ] ] Range: A row module over Z of rank 1
gap> k := WeakKernelEmbedding( proj1 ); <A morphism in Rows( Z )>
gap> Display( k ); A morphism in Rows( Z ) Source: A row module over Z of rank 1 Matrix: [ [ 0, 0, 1 ] ] Range: A row module over Z of rank 3
gap> ck := WeakCokernelProjection( k ); <A morphism in Rows( Z )>
gap> Display( ck ); A morphism in Rows( Z ) Source: A row module over Z of rank 3 Matrix: [ [ 0, -1 ], [ 1, 0 ], [ 0, 0 ] ] Range: A row module over Z of rank 2
gap> IsMonomorphism( k ); true gap> IsEpimorphism( k ); false gap> IsMonomorphism( ck ); false gap> IsEpimorphism( ck ); true gap> mor1 := CategoryOfRowsMorphism( obj5, HomalgMatrix( [[ 1 ], [ 2 ]], S ), obj6 ); <A morphism in Rows( Z )>
gap> Display( mor1 ); A morphism in Rows( Z ) Source: A row module over Z of rank 2 Matrix: [ [ 1 ], [ 2 ] ] Range: A row module over Z of rank 1
gap> mor2 := IdentityMorphism( obj6 ); <An identity morphism in Rows( Z )>
gap> Display( mor2 ); An identity morphism in Rows( Z ) Source: A row module over Z of rank 1 Matrix: [ [ 1 ] ] Range: A row module over Z of rank 1
gap> lift := Lift( mor1, mor2 ); <A morphism in Rows( Z )>
gap> Display( lift ); A morphism in Rows( Z ) Source: A row module over Z of rank 2 Matrix: [ [ 1 ], [ 2 ] ] Range: A row module over Z of rank 1
gap> source := CategoryOfRowsObject( 1, rows ); <A row module over Z of rank 1> gap> range := CategoryOfRowsObject( 2, rows ); <A row module over Z of rank 2> gap> mor := CategoryOfRowsMorphism( source, HomalgMatrix( [[ 2, 3 ]], S ), range ); <A morphism in Rows( Z )> gap> colift := Colift( mor2, mor ); <A morphism in Rows( Z )>
gap> Display( colift ); A morphism in Rows( Z ) Source: A row module over Z of rank 1 Matrix: [ [ 2, 3 ] ] Range: A row module over Z of rank 2
gap> fp := WeakBiFiberProduct( mor1, mor2 ); <A row module over Z of rank 2> gap> fp_proj := ProjectionOfBiasedWeakFiberProduct( mor1, mor2 ); <A morphism in Rows( Z )>
gap> Display( fp_proj ); A morphism in Rows( Z ) Source: A row module over Z of rank 2 Matrix: [ [ -2, 1 ], [ -1, 0 ] ] Range: A row module over Z of rank 2
gap> po := WeakBiPushout( mor, mor2 ); <A row module over Z of rank 2> gap> inj_push := InjectionOfBiasedWeakPushout( mor, mor2 ); <A morphism in Rows( Z )>
gap> Display( inj_push ); A morphism in Rows( Z ) Source: A row module over Z of rank 2 Matrix: [ [ -3, 1 ], [ 2, -1 ] ] Range: A row module over Z of rank 2
gap> Q := HomalgFieldOfRationals();; gap> rows := CategoryOfRows( Q );; gap> homalg_matrix := HomalgMatrix( [ [ 1, 0 ], > [ 0, 1 ], > [ -2, 0] ], 3, 2, Q );; gap> alpha := homalg_matrix / rows;; gap> diagram := [ alpha, 2*alpha, 3*alpha ];; gap> equalizer := EmbeddingOfEqualizer( diagram );; gap> Display(equalizer); Source: A row module over Q of rank 1 Matrix: [ [ 2, 0, 1 ] ] Range: A row module over Q of rank 3 A monomorphism in Rows( Q ) gap> equalizer_d1 := PreCompose( equalizer, diagram[1] );; gap> equalizer_d2 := PreCompose( equalizer, diagram[2] );; gap> equalizer_d3 := PreCompose( equalizer, diagram[3] );; gap> IsEqualForMorphisms( equalizer_d1, equalizer_d2 ); true gap> IsEqualForMorphisms( equalizer_d2, equalizer_d3 ); true gap> IsEqualForMorphisms( equalizer_d1, equalizer_d3 ); true gap> universal_morphism := UniversalMorphismIntoEqualizer( Source(alpha), diagram, Source( 2*equalizer ), 2*equalizer );; gap> Display( universal_morphism ); Source: A row module over Q of rank 1 Matrix: [ [ 2 ] ] Range: A row module over Q of rank 1 A morphism in Rows( Q ) gap> IsEqualForMorphisms( PreCompose( universal_morphism, equalizer ), 2*equalizer ); true
gap> Q := HomalgFieldOfRationals();; gap> rows := CategoryOfRows( Q );; gap> homalg_matrix := HomalgMatrix( [ [ 1, 0, -2 ], > [ 0, 1, 0 ] ], 2, 3, Q );; gap> alpha := homalg_matrix / rows;; gap> diagram := [ alpha, 2*alpha, 3*alpha ];; gap> coequalizer := ProjectionOntoCoequalizer( diagram );; gap> Display( coequalizer ); Source: A row module over Q of rank 3 Matrix: [ [ 2 ], [ 0 ], [ 1 ] ] Range: A row module over Q of rank 1 An epimorphism in Rows( Q ) gap> coequalizer_d1 := PreCompose( diagram[1], coequalizer );; gap> coequalizer_d2 := PreCompose( diagram[2], coequalizer );; gap> coequalizer_d3 := PreCompose( diagram[3], coequalizer );; gap> IsEqualForMorphisms( coequalizer_d1, coequalizer_d2 ); true gap> IsEqualForMorphisms( coequalizer_d2, coequalizer_d3 ); true gap> IsEqualForMorphisms( coequalizer_d1, coequalizer_d3 ); true gap> universal_morphism := UniversalMorphismFromCoequalizer( Range(alpha), diagram, Range( 2*coequalizer ), 2*coequalizer );; gap> Display( universal_morphism ); Source: A row module over Q of rank 1 Matrix: [ [ 2 ] ] Range: A row module over Q of rank 1 A morphism in Rows( Q ) gap> IsEqualForMorphisms( PreCompose( coequalizer, universal_morphism ), 2*coequalizer ); true
gap> R := HomalgRingOfIntegers();; gap> rows := CategoryOfRows( R );; gap> M := HomalgMatrix( [ [ 2, 2, 2 ], [ 3, 3, 3 ] ], 2, 3, R );; gap> alpha := AsCategoryOfRowsMorphism( M, rows );; gap> pi := PreCompose( [ > SimplifySourceAndRange_IsoFromInputSource( alpha, infinity ), > SimplifySourceAndRange( alpha, infinity ), > SimplifySourceAndRange_IsoToInputRange( alpha, infinity ) ] );; gap> IsCongruentForMorphisms( pi, alpha ); true gap> IsOne( > PreCompose( SimplifySourceAndRange_IsoFromInputSource( alpha, infinity ), SimplifySourceAndRange_IsoToInputSource( alpha, infinity ) ) > ); true gap> IsOne( > PreCompose( SimplifySourceAndRange_IsoFromInputRange( alpha, infinity ), SimplifySourceAndRange_IsoToInputRange( alpha, infinity ) ) > ); true gap> pi2 := PreCompose( > SimplifySource_IsoFromInputObject( alpha, infinity ), > SimplifySource( alpha, infinity ) > );; gap> IsCongruentForMorphisms( pi2, alpha ); true gap> IsOne( PreCompose( SimplifySource_IsoFromInputObject( alpha, infinity ), SimplifySource_IsoToInputObject( alpha, infinity ) ) ); true gap> pi3 := PreCompose( > SimplifyRange( alpha, infinity ), > SimplifyRange_IsoToInputObject( alpha, infinity ) > );; gap> IsCongruentForMorphisms( pi3, alpha ); true gap> IsOne( PreCompose( SimplifyRange_IsoFromInputObject( alpha, infinity ), SimplifyRange_IsoToInputObject( alpha, infinity ) ) ); true
gap> R := HomalgRingOfIntegers();; gap> cols := CategoryOfColumns( R );; gap> M := HomalgMatrix( [ [ 2, 2, 2 ], [ 3, 3, 3 ] ], 2, 3, R );; gap> alpha := AsCategoryOfColumnsMorphism( M, cols );; gap> pi := PreCompose( [ > SimplifySourceAndRange_IsoFromInputSource( alpha, infinity ), > SimplifySourceAndRange( alpha, infinity ), > SimplifySourceAndRange_IsoToInputRange( alpha, infinity ) ] );; gap> IsCongruentForMorphisms( pi, alpha ); true gap> IsOne( > PreCompose( SimplifySourceAndRange_IsoFromInputSource( alpha, infinity ), SimplifySourceAndRange_IsoToInputSource( alpha, infinity ) ) > ); true gap> IsOne( > PreCompose( SimplifySourceAndRange_IsoFromInputRange( alpha, infinity ), SimplifySourceAndRange_IsoToInputRange( alpha, infinity ) ) > ); true gap> pi2 := PreCompose( > SimplifySource_IsoFromInputObject( alpha, infinity ), > SimplifySource( alpha, infinity ) > );; gap> IsCongruentForMorphisms( pi2, alpha ); true gap> IsOne( PreCompose( SimplifySource_IsoFromInputObject( alpha, infinity ), SimplifySource_IsoToInputObject( alpha, infinity ) ) ); true gap> pi3 := PreCompose( > SimplifyRange( alpha, infinity ), > SimplifyRange_IsoToInputObject( alpha, infinity ) > );; gap> IsCongruentForMorphisms( pi3, alpha ); true gap> IsOne( PreCompose( SimplifyRange_IsoFromInputObject( alpha, infinity ), SimplifyRange_IsoToInputObject( alpha, infinity ) ) ); true
gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";; gap> A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );; gap> M1 := HomalgMatrix( "[ \ > Dx \ > ]", 1, 1, A3 );; gap> M2 := HomalgMatrix( "[ \ > Dx, \ > Dy \ > ]", 2, 1, A3 );; gap> M3 := HomalgMatrix( "[ \ > Dx, \ > Dy, \ > Dz \ > ]", 3, 1, A3 );; gap> M := DiagMat( [ M1, M2, M3 ] );; gap> M := ShallowCopy( M );; gap> SetIsMutableMatrix( M, true );; gap> M[ 1, 2 ] := "1";; gap> M[ 2, 3 ] := "1";; gap> M[ 3, 3 ] := "1";; gap> MakeImmutable( M );; gap> tau1 := HomalgMatrix( "[ \ > 1, Dx, Dz, \ > 0, 0, 1, \ > 0, 1, Dy \ > ]", 3, 3, A3 );; gap> tau2 := HomalgMatrix( "[ \ > 0, 1, Dz+x*y, \ > 0, 0, 1, \ > 1, Dz, x-y \ > ]", 3, 3, A3 );; gap> tau3 := HomalgMatrix( "[ \ > 1, 0, 0, \ > 1, 1, 0, \ > 0, -1, 1 \ > ]", 3, 3, A3 );; gap> tau := tau1 * tau2 * tau3;; gap> M := M * tau;; gap> rows := CategoryOfRows( A3 );; gap> alpha := AsCategoryOfRowsMorphism( M, rows );; gap> Mrows := FreydCategoryObject( alpha );; gap> Srows := SimplifyObject( Mrows, infinity );; gap> RankOfObject( Source( RelationMorphism( Srows ) ) ); 4 gap> RankOfObject( Range( RelationMorphism( Srows ) ) ); 2 gap> IsIsomorphism( SimplifyObject_IsoFromInputObject( Mrows, infinity ) ); true gap> IsIsomorphism( SimplifyObject_IsoToInputObject( Mrows, infinity ) ); true
Computing the grade filtration:
gap> mu1 := GradeFiltrationNthMonomorphism( Mrows, 1 );; gap> IsZero( mu1 ); false gap> IsMonomorphism( mu1 ); true gap> mu2 := GradeFiltrationNthMonomorphism( Mrows, 2 );; gap> IsZero( mu2 ); false gap> IsMonomorphism( mu2 ); true gap> mu3 := GradeFiltrationNthMonomorphism( Mrows, 3 );; gap> IsZero( mu3 ); false gap> IsMonomorphism( mu3 ); true gap> mu4 := GradeFiltrationNthMonomorphism( Mrows, 4 );; gap> IsZero( mu4 ); true
gap> cols := CategoryOfColumns( A3 );; gap> alpha := AsCategoryOfColumnsMorphism( M, cols );; gap> Mcols := FreydCategoryObject( alpha );; gap> Scols := SimplifyObject( Mcols, infinity );; gap> RankOfObject( Source( RelationMorphism( Scols ) ) ); 1 gap> RankOfObject( Range( RelationMorphism( Scols ) ) ); 4 gap> IsIsomorphism( SimplifyObject_IsoFromInputObject( Mcols, infinity ) ); true gap> IsIsomorphism( SimplifyObject_IsoToInputObject( Mcols, infinity ) ); true
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