Goto Chapter: Top 1 2 Ind

### 2 Examples and Tests

#### 2.1 IsZero

gap> QQ := HomalgFieldOfRationalsInSingular( );;
gap> R := QQ * "x,y";
Q[x,y]
gap> category := IntrinsicCategory( LeftPresentations( R ) );
IntrinsicCategory( Category of left presentations of Q[x,y] )
gap> M := AsLeftPresentation( HomalgMatrix( "[ 1, x, 0, 1 ]", 2, 2, R ) );
<An object in Category of left presentations of Q[x,y]>
gap> phi := CoverByFreeModule( M );
<A morphism in Category of left presentations of Q[x,y]>
gap> M := Intrinsify( category, M );
<an intrinsic object on active cell: <An object in Category
of left presentations of Q[x,y]>>
gap> F := Intrinsify( category, Source( phi ) );
<an intrinsic object on active cell:
<A projective object in Category of left presentations of Q[x,y]>>
gap> phi := Intrinsify( phi, F, 1, M, 1 );
<an intrinsic morphism on active cell:
<A morphism in Category of left presentations of Q[x,y]>>
gap> Display( phi );
1,0,
0,1

A morphism in Category of left presentations of Q[x,y]
gap> m1 := MorphismFromZeroObject( M );
<an intrinsic morphism on active cell:
<A zero, split monomorphism in Category of left presentations of Q[x,y]>>
gap> m2 := MorphismIntoZeroObject( M );
<an intrinsic morphism on active cell:
<A zero, split epimorphism in Category of left presentations of Q[x,y]>>
gap> IsZero( M );
true
gap> ActiveCell( m1 );
<A zero, isomorphism in Category of left presentations of Q[x,y]>
gap> ActiveCell( m2 );
<A zero, isomorphism in Category of left presentations of Q[x,y]>
gap> Display( phi );
(an empty 2 x 0 matrix)

A zero, split epimorphism in Category of left presentations of Q[x,y]


#### 2.2 TurnAutoequivalenceIntoIdentityFunctor

gap> QQ := HomalgFieldOfRationalsInSingular( );;
gap> R := QQ * "x,y";
Q[x,y]
gap> category := IntrinsicCategory( LeftPresentations( R ) );
IntrinsicCategory( Category of left presentations of Q[x,y] )
gap> M := AsLeftPresentation( HomalgMatrix( "[ 1, x, 0, 1 ]", 2, 2, R ) );
<An object in Category of left presentations of Q[x,y]>
gap> phi := CoverByFreeModule( M );
<A morphism in Category of left presentations of Q[x,y]>
gap> M := Intrinsify( category, M );
<an intrinsic object on active cell: <An object in Category
of left presentations of Q[x,y]>>
gap> F := Intrinsify( category, Source( phi ) );
<an intrinsic object on active cell:
<A projective object in Category of left presentations of Q[x,y]>>
gap> phi := Intrinsify( phi, F, 1, M, 1 );
<an intrinsic morphism on active cell:
<A morphism in Category of left presentations of Q[x,y]>>
gap> PositionOfActiveCell( phi );
[ 1, 1, 1 ]
gap> Id := IdentityFunctor( category );
Identity functor of
IntrinsicCategory( Category of left presentations of Q[x,y] )
gap> lg := FunctorLessGeneratorsLeft( R );
Less generators for Category of left presentations of Q[x,y]
gap> LG := Intrinsify( lg, category );
Intrinsic version of Less generators
for Category of left presentations of Q[x,y]
gap> etaLG := Intrinsify(
>   NaturalIsomorphismFromIdentityToLessGeneratorsLeft( R ), Id, LG );
Intrinsic version of Natural isomorphism
from Id to Less generators for Category of left presentations of Q[x,y]
gap> IdLG := TurnAutoequivalenceIntoIdentityFunctor( etaLG );
Intrinsic version of Less generators for
Category of left presentations of Q[x,y] as identity functor with side effects
gap> ApplyFunctor( IdLG, phi );
<an intrinsic morphism on active cell:
<A morphism in Category of left presentations of Q[x,y]>>
gap> PositionOfActiveCell( phi );
[ 1, 2, 1 ]
gap> ApplyFunctor( IdLG, phi );
<an intrinsic morphism on active cell:
<A morphism in Category of left presentations of Q[x,y]>>
gap> PositionOfActiveCell( phi );
[ 1, 2, 1 ]

gap> QQ := HomalgFieldOfRationalsInSingular( );;
gap> R := QQ * "x,y";
Q[x,y]
gap> category := IntrinsicCategory( LeftPresentations( R ) );
IntrinsicCategory( Category of left presentations of Q[x,y] )
gap> M := AsLeftPresentation( HomalgMatrix( "[ 1, x, 0, 1 ]", 2, 2, R ) );
<An object in Category of left presentations of Q[x,y]>
gap> phi := CoverByFreeModule( M );
<A morphism in Category of left presentations of Q[x,y]>
gap> M := Intrinsify( category, M );
<an intrinsic object on active cell: <An object in Category
of left presentations of Q[x,y]>>
gap> F := Intrinsify( category, Source( phi ) );
<an intrinsic object on active cell:
<A projective object in Category of left presentations of Q[x,y]>>
gap> phi := Intrinsify( phi, F, 1, M, 1 );
<an intrinsic morphism on active cell:
<A morphism in Category of left presentations of Q[x,y]>>
gap> PositionOfActiveCell( phi );
[ 1, 1, 1 ]
gap> Id := IdentityFunctor( category );
Identity functor of
IntrinsicCategory( Category of left presentations of Q[x,y] )
gap> sm := FunctorStandardModuleLeft( R );
Standard module for Category of left presentations of Q[x,y]
gap> SM := Intrinsify( sm, category );
Intrinsic version of Standard module
for Category of left presentations of Q[x,y]
gap> etaSM := Intrinsify(
>   NaturalIsomorphismFromIdentityToStandardModuleLeft( R ), Id, SM );
Intrinsic version of Natural isomorphism
from Id to Standard module for Category of left presentations of Q[x,y]
gap> IdSM := TurnAutoequivalenceIntoIdentityFunctor( etaSM );
Intrinsic version of Standard module for
Category of left presentations of Q[x,y] as identity functor with side effects
gap> ApplyFunctor( IdSM, phi );
<an intrinsic morphism on active cell:
<A morphism in Category of left presentations of Q[x,y]>>
gap> PositionOfActiveCell( phi );
[ 1, 2, 1 ]
gap> ApplyFunctor( IdSM, phi );
<an intrinsic morphism on active cell:
<A morphism in Category of left presentations of Q[x,y]>>
gap> PositionOfActiveCell( phi );
[ 1, 2, 1 ]


#### 2.3 Spectral Sequences

gap> QQ := HomalgFieldOfRationalsInSingular( );;
gap> R := QQ * "x,y";
Q[x,y]
gap> category := IntrinsicCategory( LeftPresentations( R ) );
IntrinsicCategory( Category of left presentations of Q[x,y] )
gap> S := Intrinsify( category, FreeLeftPresentation( 1, R ) );
<an intrinsic object on active cell:
<A projective object in Category of left presentations of Q[x,y]>>
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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> S2 := Intrinsify( category, FreeLeftPresentation( 2, R ) );
<an intrinsic object on active cell:
<A projective object in Category of left presentations of Q[x,y]>>
gap> C1 := ZFunctorObjectFromMorphismList( [ InjectionOfCofactorOfDirectSum( [ S2, S ], 1 ) ], 2 );
<An object in Functors from integers into
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> C1 := ZFunctorObjectExtendedByInitialAndIdentity( C1, 2, 3 );
<An object in Functors from integers into
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> C2 := ZFunctorObjectFromMorphismList( [ InjectionOfCofactorOfDirectSum( [ S, S ], 1 ) ], 3 );
<An object in Functors from integers into
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> C2 := ZFunctorObjectExtendedByInitialAndIdentity( C2, 3, 4 );
<An object in Functors from integers into
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> delta_1_3 := PresentationMorphism( ActiveCell( C1[3] ), HomalgMatrix( [ [ "x^2" ], [ "xy" ], [ "y^3"] ], 3, 1, R ), ActiveCell( C0[3] ) );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta_1_3 := Intrinsify( delta_1_3, C1[3], 1, C0[3], 1 );
<an intrinsic morphism on active cell: <A morphism in Category
of left presentations of Q[x,y]>>
gap> delta_1_2 := PresentationMorphism( ActiveCell( C1[2] ), HomalgMatrix( [ [ "x^2" ], [ "xy" ] ], 2, 1, R ), ActiveCell( C0[2] ) );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta_1_2 := Intrinsify( delta_1_2, C1[2], 1, C0[2], 1 );
<an intrinsic morphism on active cell: <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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> delta1 := ZFunctorMorphismExtendedByInitialAndIdentity( delta1, 0, 3 );
<A morphism in Functors from integers into
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> delta1 := AsAscendingFilteredMorphism( delta1 );
<A morphism in Ascending filtered object category of
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> delta_2_3 := PresentationMorphism( ActiveCell( C2[3] ), HomalgMatrix( [ [ "y", "-x", "0" ] ], 1, 3, R ), ActiveCell( C1[3] ) );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta_2_3 := Intrinsify( delta_2_3, C2[3], 1, C1[3], 1 );
<an intrinsic morphism on active cell: <A morphism in Category
of left presentations of Q[x,y]>>
gap> delta_2_4 := PresentationMorphism( ActiveCell( C2[4] ), HomalgMatrix( [ [ "y", "-x", "0" ], [ "0", "y^2", "-x" ] ], 2, 3, R ), ActiveCell( C1[4] ) );
<A morphism in Category of left presentations of Q[x,y]>
gap> delta_2_4 := Intrinsify( delta_2_4, C2[4], 1, C1[4], 1 );
<an intrinsic morphism on active cell: <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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> delta2 := ZFunctorMorphismExtendedByInitialAndIdentity( delta2, 2, 4 );
<A morphism in Functors from integers into
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> delta2 := AsAscendingFilteredMorphism( delta2 );
<A morphism in Ascending filtered object category of
IntrinsicCategory( 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 IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> complex := AsComplex( complex );
<An object in Complex category of Ascending filtered object category
of IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> LessGenFunctor := FunctorLessGeneratorsLeft( R );
Less generators for Category of left presentations of Q[x,y]
gap> Id := IdentityFunctor( category );
Identity functor of
IntrinsicCategory( Category of left presentations of Q[x,y] )
gap> LessGenFunctor := Intrinsify( LessGenFunctor, category );
Intrinsic version of Less generators
for Category of left presentations of Q[x,y]
gap> etaLG := Intrinsify(
>   NaturalIsomorphismFromIdentityToLessGeneratorsLeft( R ), Id, LessGenFunctor );
Intrinsic version of Natural isomorphism
from Id to Less generators for Category of left presentations of Q[x,y]
gap> IdLG := TurnAutoequivalenceIntoIdentityFunctor( etaLG );
Intrinsic version of Less generators for
Category of left presentations of Q[x,y] as identity functor with side effects
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 0, 0, 0 );
<A morphism in Generalized morphism category of
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( 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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( 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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( 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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) ) );
x*y,
x^2
gap> s := SpectralSequenceEntryOfAscendingFilteredComplex( complex, 4, 0, 0 );
<A morphism in Generalized morphism category of
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( 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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) ) );
x*y,
x^2,
y^3
gap> s := SpectralSequenceDifferentialOfAscendingFilteredComplex( complex, 3, 3, -2 );
<an intrinsic morphism on active cell: <A morphism in Category
of left presentations of Q[x,y]>>
gap> Display( UnderlyingMatrix( ActiveCell( ApplyFunctor( LessGenFunctor, s ) ) ) );
y^3
gap> AscToDescFunctor := AscendingToDescendingFilteredObjectFunctor( category );
Ascending to descending filtered object functor of
IntrinsicCategory( 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 IntrinsicCategory( 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 IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 0, -2, 1 );
<A morphism in Generalized morphism category of
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( 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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( 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
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) ) );
x,-y
gap> s := SpectralSequenceEntryOfDescendingFilteredCocomplex( cocomplex, 3, -2, 1 );
<A morphism in Generalized morphism category of
IntrinsicCategory( Category of left presentations of Q[x,y] )>
gap> Display( UnderlyingMatrix( ActiveCell( ApplyFunctor( LessGenFunctor, UnderlyingHonestObject( Source( s ) ) ) ) ) );
(an empty 0 x 0 matrix)
gap> s := SpectralSequenceDifferentialOfDescendingFilteredCocomplex( cocomplex, 2, -2, 1 );
<an intrinsic morphism on active cell: <A morphism in Category
of left presentations of Q[x,y]>>
gap> Display( UnderlyingMatrix( ActiveCell( ApplyFunctor( LessGenFunctor, s ) ) ) );
x*y,
x^2

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