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6 The coframe of Zariski closed subsets in an affine variety
6.1 GAP Categories
6.1-1 IsZariskiCoframeOfAnAffineVariety
‣ IsZariskiCoframeOfAnAffineVariety( object ) | ( filter ) |
Returns: true or false
The GAP category of Zariski coframes of an affine variety.
6.1-2 IsObjectInZariskiCoframeOfAnAffineVariety
‣ IsObjectInZariskiCoframeOfAnAffineVariety( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in a Zariski coframe of an affine variety.
6.1-3 IsMorphismInZariskiCoframeOfAnAffineVariety
‣ IsMorphismInZariskiCoframeOfAnAffineVariety( morphism ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a Zariski coframe of an affine variety.
6.2 Constructors
6.2-1 ZariskiCoframeOfAffineSpectrum
‣ ZariskiCoframeOfAffineSpectrum( R ) | ( attribute ) |
Returns: a CAP category
Construct the Zariski coframe of closed sets in an affine variety defined as the vanishing loci of (radical) ideals of a homalg ring R.
6.2-2 ClosedSubsetOfSpec
‣ ClosedSubsetOfSpec( mat ) | ( operation ) |
‣ ClosedSubsetOfSpec( str, R ) | ( operation ) |
‣ ClosedSubsetOfSpec( r ) | ( operation ) |
‣ ClosedSubsetOfSpecByListOfColumns( L ) | ( operation ) |
‣ ClosedSubsetOfSpecByListOfRadicalColumns( L ) | ( operation ) |
Returns: a CAP object
Construct a Zariski closed subset (as an object in the Zariski coframe of closed subsets in an affine variety) from a matrix mat. The result is the 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 vanishing locus of the annihilator of \(M\).
gap> ZZ := HomalgRingOfIntegers( );
Z
gap> ZC := ZariskiCoframeOfAffineSpectrum( ZZ );
The coframe of Zariski closed subsets of the affine spectrum of Z
gap> A := ClosedSubsetOfSpec( HomalgMatrix( [ 4 ], 1, 1, ZZ ) );
V_{Z}( <...> )
gap> Display( A );
V( <4> )
gap> B := ClosedSubsetOfSpec( "[ 12, 20 ]", ZZ );
V_{Z}( <...> )
gap> C := ClosedSubsetOfSpecByRadicalColumn( "[ 3 ]", ZZ );
V_{Z}( <...> )
gap> D := ClosedSubsetOfSpec( "[ 12 ]", ZZ );
V_{Z}( <...> )
gap> A = B;
true
gap> Display( A );
V( <2> )
gap> A = C;
false
gap> LCA := LocallyClosedPart( A );
V_{Z}( <...> )
gap> Display( LCA );
V( <2> )
gap> A = LCA;
true
gap> embAB := UniqueMorphism( A, B );
<An epi-, monomorphism in The coframe of Zariski closed 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 coframe of Zariski closed subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAC );
false
gap> embAD := UniqueMorphism( A, D );
<An epi-, monomorphism in The coframe of Zariski closed subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embAD );
true
gap> IsSubset( D, A );
true
gap> IsIsomorphism( embAD );
false
gap> embDA := UniqueMorphism( D, A );
<An epi-, monomorphism in The coframe of Zariski closed subsets of the\
affine spectrum of Z>
gap> IsWellDefined( embDA );
false
gap> IsSubset( A, D );
false
gap> T := TerminalObject( ZC );
V_{Z}( <...> )
gap> Display( T );
V( <> )
gap> I := InitialObject( ZC );
V_{Z}( <...> )
gap> Display( I );
∅
gap> A := ClosedSubsetOfSpec( "[ 4 ]", ZZ );
V_{Z}( <...> )
gap> Display( A );
V( <4> )
gap> AvC := Coproduct( A, C );
V_{Z}( <...> )
gap> Display( AvC );
{ V( <4> ) ∪ V( <3> ) }
gap> StandardizeObject( AvC );
V_{Z}( <...> )
gap> Display( AvC );
{ V( <2> ) ∪ V( <3> ) }
gap> AC := DirectProduct( A, C );
V_{Z}( <...> )
gap> Display( AC );
V( <4,3> )
gap> StandardizeObject( AC );
V_{Z}( <...> )
gap> Display( AC );
∅
gap> DirectProduct( A, C ) = I;
true
gap> CoexponentialOnObjects( I, T ) = I;
true
gap> CoexponentialOnObjects( I, A ) = I;
true
gap> CoexponentialOnObjects( I, I ) = I;
true
gap> CoexponentialOnObjects( A, I ) = A;
true
gap> CoexponentialOnObjects( T, I ) = T;
true
gap> CoexponentialOnObjects( T, A ) = T;
true
gap> CoexponentialOnObjects( T, T ) = I;
true
gap> CoexponentialOnObjects( B, D ) = I;
true
gap> CoexponentialOnObjects( D, B ) = C;
true
gap> CoexponentialOnObjects( C, D ) = I;
true
gap> CoexponentialOnObjects( D, C ) = B;
true
gap> p := AClosedSingleton( D );
V_{Z}( <...> )
gap> Display( p );
V( <2> )
gap> q := AClosedSingleton( D - p );
V_{Z}( <...> )
gap> Display( q );
V( <3> )
gap> p + q = D;
true
6.2-3 ClosedSubsetOfSpecByRadicalColumn
‣ ClosedSubsetOfSpecByRadicalColumn( I ) | ( operation ) |
‣ ClosedSubsetOfSpecByRadicalColumn( str, R ) | ( operation ) |
‣ ClosedSubsetOfSpecByRadicalColumn( r ) | ( operation ) |
ClosedSubsetOfSpecByRadicalColumn assumes that the image is a radical ideal.
6.2-4 ClosedSubsetOfSpecByStandardColumn
‣ ClosedSubsetOfSpecByStandardColumn( I ) | ( operation ) |
‣ ClosedSubsetOfSpecByStandardColumn( str, R ) | ( operation ) |
‣ ClosedSubsetOfSpecByStandardColumn( r ) | ( operation ) |
ClosedSubsetOfSpecByStandardColumn assumes that the image is a radical ideal given by some sort of a "standard" basis.
6.2-5 TangentSpaceAtPoint
‣ TangentSpaceAtPoint( V, p ) | ( operation ) |
Returns: an object in a Zariski coframe
Compute the tangent space of V at the closed point p as an affine subspace of the ambient space of V intersecting p.
6.2-6 TangentSpaceAtPoint
‣ TangentSpaceAtPoint( V, p ) | ( operation ) |