Examples and Tests
Annihilator
gap> ZZ := HomalgRingOfIntegersInSingular();; gap> M1 := AsLeftPresentation( HomalgMatrix( [ [ "2" ] ], ZZ ) );; gap> M2 := AsLeftPresentation( HomalgMatrix( [ [ "3" ] ], ZZ ) );; gap> M3 := AsLeftPresentation( HomalgMatrix( [ [ "4" ] ], ZZ ) );; gap> M := DirectSum( M1, M2, M3 );; gap> Display( Annihilator( M ) ); 12 A monomorphism in Category of left presentations of Z gap> M1 := AsRightPresentation( HomalgMatrix( [ [ "2" ] ], ZZ ) );; gap> M2 := AsRightPresentation( HomalgMatrix( [ [ "3" ] ], ZZ ) );; gap> M3 := AsRightPresentation( HomalgMatrix( [ [ "4" ] ], ZZ ) );; gap> M := DirectSum( M1, M2, M3 );; gap> Display( Annihilator( M ) ); 12 A monomorphism in Category of right presentations of Z
Intersection of Submodules
gap> Q := HomalgFieldOfRationalsInSingular();; gap> R := Q * "x,y"; Q[x,y] gap> F := AsLeftPresentation( HomalgMatrix( [ [ 0 ] ], R ) ); <An object in Category of left presentations of Q[x,y]> gap> I1 := AsLeftPresentation( HomalgMatrix( [ [ "x" ] ], R ) );; gap> I2 := AsLeftPresentation( HomalgMatrix( [ [ "y" ] ], R ) );; gap> Display( I1 ); x An object in Category of left presentations of Q[x,y] gap> Display( I2 ); y An object in Category of left presentations of Q[x,y] gap> eps1 := PresentationMorphism( F, HomalgMatrix( [ [ 1 ] ], R ), I1 ); <A morphism in Category of left presentations of Q[x,y]> gap> eps2 := PresentationMorphism( F, HomalgMatrix( [ [ 1 ] ], R ), I2 ); <A morphism in Category of left presentations of Q[x,y]> gap> kernelemb1 := KernelEmbedding( eps1 ); <A monomorphism in Category of left presentations of Q[x,y]> gap> kernelemb2 := KernelEmbedding( eps2 ); <A monomorphism in Category of left presentations of Q[x,y]> gap> P := FiberProduct( kernelemb1, kernelemb2 );; gap> Display( P ); (an empty 0 x 1 matrix) An object in Category of left presentations of Q[x,y] gap> pi1 := ProjectionInFactorOfFiberProduct( [ kernelemb1, kernelemb2 ], 1 ); <A monomorphism in Category of left presentations of Q[x,y]> gap> composite := PreCompose( pi1, kernelemb1 ); <A monomorphism in Category of left presentations of Q[x,y]> gap> Display( composite ); x*y A monomorphism in Category of left presentations of Q[x,y]
Koszul Complex
gap> Q := HomalgFieldOfRationalsInSingular();; gap> R := Q * "x,y,z";; gap> M := HomalgMatrix( [ [ "x" ], [ "y" ], [ "z" ] ], 3, 1, R );; gap> Ml := AsLeftPresentation( M );; gap> eps := CoverByFreeModule( Ml );; gap> iota1 := KernelEmbedding( eps );; gap> Display( iota1 ); x, y, z A monomorphism in Category of left presentations of Q[x,y,z] gap> Display( Source( iota1 ) ); 0, -z,y, -z,0, x, -y,x, 0 An object in Category of left presentations of Q[x,y,z] gap> pi1 := CoverByFreeModule( Source( iota1 ) );; gap> d1 := PreCompose( pi1, iota1 );; gap> Display( d1 ); x, y, z A morphism in Category of left presentations of Q[x,y,z] gap> iota2 := KernelEmbedding( d1 );; gap> Display( iota2 ); 0, -z,y, -z,0, x, -y,x, 0 A monomorphism in Category of left presentations of Q[x,y,z] gap> Display( Source( iota2 ) );; x,-y,z An object in Category of left presentations of Q[x,y,z] gap> pi2 := CoverByFreeModule( Source( iota2 ) );; gap> d2 := PreCompose( pi2, iota2 );; gap> Display( d2 ); 0, -z,y, -z,0, x, -y,x, 0 A morphism in Category of left presentations of Q[x,y,z] gap> iota3 := KernelEmbedding( d2 );; gap> Display( iota3 ); x,-y,z A monomorphism in Category of left presentations of Q[x,y,z] gap> Display( Source( iota3 ) ); (an empty 0 x 1 matrix) An object in Category of left presentations of Q[x,y,z] gap> pi3 := CoverByFreeModule( Source( iota3 ) );; gap> d3 := PreCompose( pi3, iota3 );; gap> Display( d3 ); x,-y,z A morphism in Category of left presentations of Q[x,y,z] gap> N := HomalgMatrix( [ [ "x" ] ], 1, 1, R );; gap> Nl := AsLeftPresentation( N );; gap> d2Nl := TensorProductOnMorphisms( d2, IdentityMorphism( Nl ) );; gap> d1Nl := TensorProductOnMorphisms( d1, IdentityMorphism( Nl ) );; gap> IsZero( PreCompose( d2Nl, d1Nl ) ); true gap> cycles := KernelEmbedding( d1Nl );; gap> boundaries := ImageEmbedding( d2Nl );; gap> boundaries_in_cyles := LiftAlongMonomorphism( cycles, boundaries );; gap> homology := CokernelObject( boundaries_in_cyles );; gap> LessGenFunctor := FunctorLessGeneratorsLeft( R );; gap> homology := ApplyFunctor( LessGenFunctor, homology );; gap> StdBasisFunctor := FunctorStandardModuleLeft( R );; gap> homology := ApplyFunctor( StdBasisFunctor, homology );; gap> Display( homology ); z, y, x An object in Category of left presentations of Q[x,y,z]
Closed Monoidal Structure
gap> R := HomalgRingOfIntegers( );; gap> M := AsLeftPresentation( HomalgMatrix( [ [ 2 ] ], 1, 1, R ) ); <An object in Category of left presentations of Z> gap> N := AsLeftPresentation( HomalgMatrix( [ [ 3 ] ], 1, 1, R ) ); <An object in Category of left presentations of Z> gap> T := TensorProductOnObjects( M, N ); <An object in Category of left presentations of Z> gap> Display( T ); [ [ 3 ], [ 2 ] ] An object in Category of left presentations of Z gap> IsZero( T ); true gap> H := InternalHomOnObjects( DirectSum( M, M ), DirectSum( M, N ) ); <An object in Category of left presentations of Z> gap> Display( H ); [ [ 0, 0, 0, -2 ], [ 1, 2, 0, 0 ], [ 0, 2, 2, 0 ], [ 2, 3, 0, 2 ] ] An object in Category of left presentations of Z gap> alpha := StandardGeneratorMorphism( H, 3 ); <A morphism in Category of left presentations of Z> gap> l := LambdaElimination( DirectSum( M, M ), DirectSum( M, N ), alpha ); <A morphism in Category of left presentations of Z> gap> IsZero( l ); false gap> Display( l ); [ [ -2, 6 ], [ -1, -3 ] ] A morphism in Category of left presentations of Z
Projectivity test
gap> Q := HomalgFieldOfRationalsInSingular();; gap> R := Q * "x";; gap> F := FreeLeftPresentation( 2, Q );; gap> HasIsProjective( F ) and IsProjective( F ); true gap> G := FreeRightPresentation( 2, Q );; gap> HasIsProjective( G ) and IsProjective( G ); true gap> M := AsLeftPresentation( HomalgMatrix( "[ x, x ]", 1, 2, R ) );; gap> IsProjective( M ); false gap> N := AsLeftPresentation( HomalgMatrix( "[ 1, x ]", 1, 2, R ) );; gap> IsProjective( N ); true