gap> ZZZ := HomalgRingOfIntegersInSingular( );
Z
gap> C1 := FreeLeftPresentation( 1, ZZZ );;
gap> C2 := FreeLeftPresentation( 2, ZZZ );;
gap> h1 := PresentationMorphism( C2, HomalgMatrix( [ [ 0 ], [ 4 ] ], ZZZ ), C1 );
<A morphism in Category of left presentations of Z>
gap> h2 := PresentationMorphism( C2, HomalgMatrix( [ [ 0 ], [ 2 ] ], ZZZ ), C1 );
<A morphism in Category of left presentations of Z>
gap> v1 := PresentationMorphism( C2, HomalgMatrix( [ [ 2, 0 ], [ 1, 2 ] ], ZZZ ), C2 );
<A morphism in Category of left presentations of Z>
gap> v2 := PresentationMorphism( C1, HomalgMatrix( [ [ 4 ] ], ZZZ ), C1 );
<A morphism in Category of left presentations of Z>
gap> cocomplex_h1 := CocomplexFromMorphismList( [ h1 ] );
<An object in Cocomplex category of Category of left presentations of Z>
gap> cocomplex_h2 := CocomplexFromMorphismList( [ h2 ] );
<An object in Cocomplex category of Category of left presentations of Z>
gap> cocomplex_mor := CochainMap( cocomplex_h2, [ v1, v2 ], cocomplex_h1 );
<A morphism in Cocomplex category of Category of left presentations of Z>
gap> Zmod := CapCategory( C1 );
Category of left presentations of Z
gap> CH0 := CohomologyFunctor( Zmod, 0 );
0-th cohomology functor of Category of left presentations of Z
gap> cmor0 := ApplyFunctor( CH0, cocomplex_mor );
<A morphism in Category of left presentations of Z>
gap> Display( UnderlyingMatrix( cmor0 ) );
2
gap> CH1 := CohomologyFunctor( Zmod, 1 );
1-th cohomology functor of Category of left presentations of Z
gap> cmor1 := ApplyFunctor( CH1, cocomplex_mor );
<A morphism in Category of left presentations of Z>
gap> Display( UnderlyingMatrix( cmor1 ) );
4
gap> ToComplex := CocomplexToComplexFunctor( Zmod );
Cocomplex to complex functor of Category of left presentations of Z
gap> complex_mor := ApplyFunctor( ToComplex, cocomplex_mor );
<A morphism in Complex category of Category of left presentations of Z>
gap> H0 := HomologyFunctor( Zmod, 0 );
0-th homology functor of Category of left presentations of Z
gap> mor0 := ApplyFunctor( H0, complex_mor );
<A morphism in Category of left presentations of Z>
gap> Display( UnderlyingMatrix( mor0 ) );
2
gap> Hm1 := HomologyFunctor( Zmod, -1 );
-1-th homology functor of Category of left presentations of Z
gap> mor1 := ApplyFunctor( Hm1, complex_mor );
<A morphism in Category of left presentations of Z>
gap> Display( UnderlyingMatrix( mor1 ) );
4

gap> QQ := HomalgFieldOfRationalsInSingular( );;
gap> R := QQ * "x,y";
Q[x,y]
gap> SetRecursionTrapInterval( 10000 );
gap> category := LeftPresentations( R );
Category of left presentations of Q[x,y]
gap> S := FreeLeftPresentation( 1, R );;
gap> object_func := function( i ) return S; end;
function( i ) ... end
gap> morphism_func := function( i ) return IdentityMorphism( S ); end;
function( i ) ... end
gap> C0 := ZFunctorObjectExtendedByInitialAndIdentity( object_func, morphism_func, category, 0, 4 );
<An object in Functors from integers into Category of left presentations of Q[x,y]>
gap> S2 := FreeLeftPresentation( 2, R );;
gap> C1 := ZFunctorObjectFromMorphismList( [ InjectionOfCofactorOfDirectSum( [ S2, S ], 1 ) ], 2 );
<An object in Functors from integers into Category of left presentations of Q[x,y]>
gap> C1 := ZFunctorObjectExtendedByInitialAndIdentity( C1, 2, 3 );
<An object in Functors from integers into Category of left presentations of Q[x,y]>
gap> C2 := ZFunctorObjectFromMorphismList( [ InjectionOfCofactorOfDirectSum( [ S, S ], 1 ) ], 3 );
<An object in Functors from integers into Category of left presentations of Q[x,y]>
gap> C2 := ZFunctorObjectExtendedByInitialAndIdentity( C2, 3, 4 );
<An object in Functors from integers into Category of left presentations of Q[x,y]>
gap> delta_1_3 := PresentationMorphism( C1[3], HomalgMatrix( [ [ "x^2" ], [ "xy" ], [ "y^3"] ], 3, 1, R ), C0[3] );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta_1_2 := PresentationMorphism( C1[2], HomalgMatrix( [ [ "x^2" ], [ "xy" ] ], 2, 1, R ), C0[2] );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta1 := ZFunctorMorphism( C1, [ UniversalMorphismFromInitialObject( C0[1] ), UniversalMorphismFromInitialObject( C0[1] ), delta_1_2, delta_1_3 ], 0, C0 );
<A morphism in Functors from integers into Category of left presentations of Q[x,y]>
gap> delta1 := ZFunctorMorphismExtendedByInitialAndIdentity( delta1, 0, 3 );
<A morphism in Functors from integers into Category of left presentations of Q[x,y]>
gap> delta1 := AsAscendingFilteredMorphism( delta1 );
<A morphism in Ascending filtered object category of Category of left presentations of Q[x,y]>
gap> delta_2_3 := PresentationMorphism( C2[3], HomalgMatrix( [ [ "y", "-x", "0" ] ], 1, 3, R ), C1[3] );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta_2_4 := PresentationMorphism( C2[4], HomalgMatrix( [ [ "y", "-x", "0" ], [ "0", "y^2", "-x" ] ], 2, 3, R ), C1[4] );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta2 := ZFunctorMorphism( C2, [  UniversalMorphismFromInitialObject( C1[2] ), delta_2_3, delta_2_4 ], 2, C1 );
<A morphism in Functors from integers into Category of left presentations of Q[x,y]>
gap> delta2 := ZFunctorMorphismExtendedByInitialAndIdentity( delta2, 2, 4 );
<A morphism in Functors from integers into Category of left presentations of Q[x,y]>
gap> delta2 := AsAscendingFilteredMorphism( delta2 );
<A morphism in Ascending filtered object category of Category of left presentations of Q[x,y]>
gap> SetIsAdditiveCategory( CategoryOfAscendingFilteredObjects( category ), true );
gap> complex := ZFunctorObjectFromMorphismList( [ delta2, delta1 ], -2 );
<An object in Functors from integers into Ascending filtered object category of Category of left presentations of Q[x,y]>
gap> complex := AsComplex( complex );
<An object in Complex category of Ascending filtered object category of Category of left presentations of Q[x,y]>
gap> LessGenFunctor := FunctorLessGeneratorsLeft( R );
Less generators for Category of left presentations of Q[x,y]
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 0, 0, 0 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
(an empty 0 x 1 matrix)
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 1, 0, 0 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
(an empty 0 x 1 matrix)
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 2, 0, 0 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
(an empty 0 x 1 matrix)
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 3, 0, 0 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
x*y,
x^2
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 4, 0, 0 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
x*y,
x^2,
y^3
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 5, 0, 0 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
x*y,
x^2,
y^3
gap> s := SpectralSequenceDifferentialOfAscendingFilteredComplex( complex, 3, 3, -2 );
<A morphism in Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, s ) ) );
y^3
gap> AscToDescFunctor := AscendingToDescendingFilteredObjectFunctor( category );
Ascending to descending filtered object functor of Category of left presentations of Q[x,y]
gap> cocomplex := ZFunctorObjectFromMorphismList( [ ApplyFunctor( AscToDescFunctor, delta2 ), ApplyFunctor( AscToDescFunctor, delta1 ) ], -2 );
<An object in Functors from integers into Descending filtered object category of Category of left presentations of Q[x,y]>
gap> SetIsAdditiveCategory( CategoryOfDescendingFilteredObjects( category ), true );
gap> cocomplex := AsCocomplex( cocomplex );
<An object in Cocomplex category of Descending filtered object category of Category of left presentations of Q[x,y]>
gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 0, -2, 1 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
(an empty 0 x 2 matrix)
gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 1, -2, 1 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
(an empty 0 x 2 matrix)
gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 2, -2, 1 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
-y,x
gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 3, -2, 1 );
<A morphism in Generalized morphism category of Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) );
(an empty 0 x 0 matrix)
gap> s := SpectralSequenceDifferentialOfDescendingFilteredCocomplex( cocomplex, 2, -2, 1 );
<A morphism in Category of left presentations of Q[x,y]>
gap> Display( UnderlyingMatrix( ApplyFunctor( LessGenFunctor, s ) ) );
x^2,
x*y