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5 The frame of Zariski open subsets in an affine variety
5.1 GAP Categories
5.1-1 IsZariskiFrameOfAnAffineVariety
‣ IsZariskiFrameOfAnAffineVariety( object ) | ( filter ) |
Returns: true or false
The GAP category of Zariski frames of an affine variety.
5.1-2 IsObjectInZariskiFrameOfAnAffineVariety
‣ IsObjectInZariskiFrameOfAnAffineVariety( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in a Zariski frame of an affine variety.
5.1-3 IsMorphismInZariskiFrameOfAnAffineVariety
‣ IsMorphismInZariskiFrameOfAnAffineVariety( morphism ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a Zariski frame of an affine variety.
5.2 Constructors
5.2-1 ZariskiFrameOfAffineSpectrum
‣ ZariskiFrameOfAffineSpectrum( R ) | ( attribute ) |
Returns: a CAP category
Construct the Zariski frame of open sets in an affine variety defined as the complements of vanishing loci of (radical) ideals of a homalg ring R.
5.2-2 OpenSubsetOfSpec
‣ OpenSubsetOfSpec( mat ) | ( operation ) |
‣ OpenSubsetOfSpec( str, R ) | ( operation ) |
‣ OpenSubsetOfSpec( r ) | ( operation ) |
‣ OpenSubsetOfSpecByListOfColumns( L ) | ( operation ) |
Returns: a CAP object
Construct a Zariski open subset (as an object in the Zariski frame of open subsets in an affine variety) from a matrix mat. The result is the non-support of the module-theoretic cokernel \(M\) of mat viewed as a morphism in the Freyd category of the associated category of rows, i.e., the result is the complement of the vanishing locus of the annihilator of \(M\).
gap> ZZ := HomalgRingOfIntegersInSingular( );
Z
gap> ZF := ZariskiFrameOfAffineSpectrum( ZZ );
The frame of Zariski open subsets of the affine spectrum of Z
gap> A := OpenSubsetOfSpec( HomalgMatrix( [ 4 ], 1, 1, ZZ ) );
D_{Z}( <...> )
gap> Display( A );
D( <4> )
gap> B := OpenSubsetOfSpec( "[ 12,20 ]", ZZ );
D_{Z}( <...> )
gap> C := OpenSubsetOfSpecByRadicalColumn( "[ 3 ]", ZZ );
D_{Z}( <...> )
gap> D := OpenSubsetOfSpec( "[ 12 ]", ZZ );
D_{Z}( <...> )
gap> A = B;
true
gap> Display( A );
D( <2> )
gap> A = C;
false
gap> LCA := LocallyClosedPart( A );
V_{Z}( I ) \ V_{Z}( J )
gap> Display( LCA );
V( <> ) \ V( <2> )
gap> embAB := UniqueMorphism( A, B );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAB );
true
gap> IsIsomorphism( embAB );
true
gap> embAC := UniqueMorphism( A, C );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAC );
false
gap> embDA := UniqueMorphism( D, A );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embDA );
true
gap> IsSubset( A, D );
true
gap> IsIsomorphism( embDA );
false
gap> embAD := UniqueMorphism( A, D );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAD );
false
gap> IsSubset( D, A );
false
gap> T := TerminalObject( ZF );
D_{Z}( <...> )
gap> Display( T );
D( <1> )
gap> I := InitialObject( ZF );
D_{Z}( <...> )
gap> Display( I );
∅
gap> A := OpenSubsetOfSpec( "[ 4 ]", ZZ );
D_{Z}( <...> )
gap> Display( A );
D( <4> )
gap> AvC := Coproduct( A, C );
D_{Z}( <...> )
gap> Display( AvC );
D( <4,3> )
gap> AvC = T;
true
gap> Display( AvC );
D( <1> )
gap> AC := DirectProduct( A, C );
D_{Z}( <...> )
gap> Display( AC );
{ D( <4> ) ∩ D( <3> ) }
gap> StandardizeObject( AC );
D_{Z}( <...> )
gap> Display( AC );
{ D( <2> ) ∩ D( <3> ) }
gap> ExponentialOnObjects( I, T ) = T;
true
gap> ExponentialOnObjects( A, T ) = T;
true
gap> ExponentialOnObjects( T, T ) = T;
true
gap> ExponentialOnObjects( T, A ) = A;
true
gap> ExponentialOnObjects( T, I ) = I;
true
gap> ExponentialOnObjects( A, I ) = I;
true
gap> ExponentialOnObjects( I, I ) = T;
true
gap> ExponentialOnObjects( D, B ) = T;
true
gap> ExponentialOnObjects( B, D ) = C;
true
gap> ExponentialOnObjects( D, C ) = T;
true
gap> ExponentialOnObjects( C, D ) = B;
true
gap> IsWellDefined( CartesianEvaluationMorphism( A, C ) );
true
gap> IsIsomorphism( CartesianEvaluationMorphism( A, C ) );
false
gap> IsWellDefined( CartesianEvaluationMorphism( C, A ) );
true
gap> IsIsomorphism( CartesianEvaluationMorphism( C, A ) );
false
gap> IsWellDefined( CartesianCoevaluationMorphism( A, D ) );
true
gap> IsIsomorphism( CartesianCoevaluationMorphism( A, D ) );
false
gap> IsWellDefined( CartesianCoevaluationMorphism( D, A ) );
true
gap> IsIsomorphism( CartesianCoevaluationMorphism( D, A ) );
false
gap> th := UniqueMorphism( DirectProduct( A, C ), C );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( th );
true
gap> th := DirectProductToExponentialAdjunctionMap( A, C, th );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( th );
true
gap> IsIsomorphism( th );
false
gap> ht := UniqueMorphism( A, ExponentialOnObjects( D, B ) );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( ht );
true
gap> ht := ExponentialToDirectProductAdjunctionMap( D, B, ht );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( ht );
true
gap> IsIsomorphism( ht );
false
gap> IsWellDefined( CartesianPreComposeMorphism( A, C, D ) );
true
gap> IsIsomorphism( CartesianPreComposeMorphism( A, C, D ) );
false
gap> p := AClosedSingleton( D );
V_{Z}( <...> )
gap> Display( p );
V( <5> )
gap> q := AClosedSingleton( D - p );
V_{Z}( <...> )
gap> Display( q );
V( <7> )
gap> p + q = D;
false
gap> ZZ := HomalgRingOfIntegersInSingular( );
Z
gap> ZF := ZariskiFrameOfAffineSpectrum( ZZ );
The frame of Zariski open subsets of the affine spectrum of Z
gap> A := OpenSubsetOfSpec( HomalgMatrix( [ 4 ], 1, 1, ZZ ) );
D_{Z}( <...> )
gap> Display( A );
D( <4> )
gap> B := OpenSubsetOfSpec( "[ 12,20 ]", ZZ );
D_{Z}( <...> )
gap> C := OpenSubsetOfSpecByRadicalColumn( "[ 3 ]", ZZ );
D_{Z}( <...> )
gap> D := OpenSubsetOfSpec( "[ 12 ]", ZZ );
D_{Z}( <...> )
gap> A = B;
true
gap> Display( A );
D( <2> )
gap> A = C;
false
gap> LCA := LocallyClosedPart( A );
V_{Z}( I ) \ V_{Z}( J )
gap> Display( LCA );
V( <> ) \ V( <2> )
gap> embAB := UniqueMorphism( A, B );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAB );
true
gap> IsIsomorphism( embAB );
true
gap> embAC := UniqueMorphism( A, C );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAC );
false
gap> embDA := UniqueMorphism( D, A );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embDA );
true
gap> IsSubset( A, D );
true
gap> IsIsomorphism( embDA );
false
gap> embAD := UniqueMorphism( A, D );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAD );
false
gap> IsSubset( D, A );
false
gap> T := TerminalObject( ZF );
D_{Z}( <...> )
gap> Display( T );
D( <1> )
gap> I := InitialObject( ZF );
D_{Z}( <...> )
gap> Display( I );
∅
gap> A := OpenSubsetOfSpec( "[ 4 ]", ZZ );
D_{Z}( <...> )
gap> Display( A );
D( <4> )
gap> AvC := Coproduct( A, C );
D_{Z}( <...> )
gap> Display( AvC );
D( <4,3> )
gap> AvC = T;
true
gap> Display( AvC );
D( <1> )
gap> AC := DirectProduct( A, C );
D_{Z}( <...> )
gap> Display( AC );
{ D( <4> ) ∩ D( <3> ) }
gap> StandardizeObject( AC );
D_{Z}( <...> )
gap> Display( AC );
{ D( <2> ) ∩ D( <3> ) }
gap> ExponentialOnObjects( I, T ) = T;
true
gap> ExponentialOnObjects( A, T ) = T;
true
gap> ExponentialOnObjects( T, T ) = T;
true
gap> ExponentialOnObjects( T, A ) = A;
true
gap> ExponentialOnObjects( T, I ) = I;
true
gap> ExponentialOnObjects( A, I ) = I;
true
gap> ExponentialOnObjects( I, I ) = T;
true
gap> ExponentialOnObjects( D, B ) = T;
true
gap> ExponentialOnObjects( B, D ) = C;
true
gap> ExponentialOnObjects( D, C ) = T;
true
gap> ExponentialOnObjects( C, D ) = B;
true
gap> IsWellDefined( CartesianEvaluationMorphism( A, C ) );
true
gap> IsIsomorphism( CartesianEvaluationMorphism( A, C ) );
false
gap> IsWellDefined( CartesianEvaluationMorphism( C, A ) );
true
gap> IsIsomorphism( CartesianEvaluationMorphism( C, A ) );
false
gap> IsWellDefined( CartesianCoevaluationMorphism( A, D ) );
true
gap> IsIsomorphism( CartesianCoevaluationMorphism( A, D ) );
false
gap> IsWellDefined( CartesianCoevaluationMorphism( D, A ) );
true
gap> IsIsomorphism( CartesianCoevaluationMorphism( D, A ) );
false
gap> th := UniqueMorphism( DirectProduct( A, C ), C );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( th );
true
gap> th := DirectProductToExponentialAdjunctionMap( A, C, th );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( th );
true
gap> IsIsomorphism( th );
false
gap> ht := UniqueMorphism( A, ExponentialOnObjects( D, B ) );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( ht );
true
gap> ht := ExponentialToDirectProductAdjunctionMap( D, B, ht );
<An epi-, monomorphism in The frame of Zariski open subsets of the\
affine spectrum of Z>
gap> IsWellDefined( ht );
true
gap> IsIsomorphism( ht );
false
gap> IsWellDefined( CartesianPreComposeMorphism( A, C, D ) );
true
gap> IsIsomorphism( CartesianPreComposeMorphism( A, C, D ) );
false
gap> p := AClosedSingleton( D );
V_{Z}( <...> )
gap> Display( p );
V( <5> )
gap> q := AClosedSingleton( D - p );
V_{Z}( <...> )
gap> Display( q );
V( <7> )
gap> p + q = D;
false
5.2-3 OpenSubsetOfSpecByRadicalColumn
‣ OpenSubsetOfSpecByRadicalColumn( I ) | ( operation ) |
‣ OpenSubsetOfSpecByRadicalColumn( str, R ) | ( operation ) |
‣ OpenSubsetOfSpecByRadicalColumn( r ) | ( operation ) |
‣ OpenSubsetOfSpecByListOfRadicalColumns( L ) | ( operation ) |
OpenSubsetOfSpecByRadicalColumn assumes that the image is a radical ideal.
5.2-4 OpenSubsetOfSpecByStandardColumn
‣ OpenSubsetOfSpecByStandardColumn( I ) | ( operation ) |
‣ OpenSubsetOfSpecByStandardColumn( str, R ) | ( operation ) |
‣ OpenSubsetOfSpecByStandardColumn( r ) | ( operation ) |
OpenSubsetOfSpecByStandardColumn assumes that the image is a radical ideal given by some sort of a "standard" basis.