‣ InfoZariskiFrames | ( info class ) |
‣ IsZariskiFrameOrCoframe( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in a Zariski frame or coframe.
‣ IsObjectInZariskiFrameOrCoframe( object ) | ( filter ) |
Returns: true or false
The GAP category of objects in a Zariski frame or coframe.
‣ IsMorphismInZariskiFrameOrCoframe( morphism ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in a Zariski frame or coframe.
‣ IsOpen( arg ) | ( property ) |
Returns: true or false
‣ IsClosed( arg ) | ( operation ) |
‣ UnderlyingRing( arg ) | ( attribute ) |
‣ BaseOfFibration( arg ) | ( attribute ) |
‣ BaseOfFibration( arg ) | ( attribute ) |
‣ Closure( A ) | ( attribute ) |
Returns: an object in the Zariski coframe
The closure of A.
‣ ParametrizedObject( A ) | ( attribute ) |
‣ ParametrizedObject( A ) | ( attribute ) |
‣ ParametrizedObject( A ) | ( attribute ) |
‣ LocallyClosedPart( A ) | ( attribute ) |
‣ DistinguishedLocallyClosedPart( A ) | ( attribute ) |
‣ DistinguishedLocallyClosedPart( A ) | ( attribute ) |
‣ DistinguishedLocallyClosedPart( A ) | ( attribute ) |
‣ AffineApproximation( A ) | ( attribute ) |
‣ CanonicalObject( A ) | ( attribute ) |
‣ RabinowitschCover( A ) | ( attribute ) |
Returns: a CAP object
The argument A is a locally closed subset of some ambient space \(X\). The result is a closed subset of the ambient space \(X \times \mathbb{A}^1 \to X\) which (set-theoretically) projects onto A with finite fibers.
‣ AClosedSuperset( A ) | ( operation ) |
Returns: an object in the Zariski coframe
Return a closed superset of A, i.e., a set which includes Closure( A ). If HasClosure( A ) = true then Closure( A ) is returned.
‣ BestUnderlyingColumn( A ) | ( operation ) |
Returns: a homalg matrix
The column matrix of generators of the defining ideal of A.
‣ NormalizeObject( A ) | ( operation ) |
Returns: the object in the Zariski frame or coframe A
‣ StandardizeObject( A ) | ( operation ) |
Returns: the object in the Zariski frame or coframe A
‣ IsSubset( A, B ) | ( operation ) |
Returns: true or false
Check if A is bigger than B w.r.t. inclusion.
‣ RingEpimorphismOfClosedSuperset( A ) | ( operation ) |
Returns: a homalg ring map
Return the ring epimorphism from the coordinate ring of the ambient space of A onto the coordinate ring of the closure of A in its ambient space.
‣ RingEpimorphismOfClosure( A ) | ( operation ) |
Returns: a homalg ring map
Return the ring epimorphism from the coordinate ring of the ambient space of A onto the coordinate ring of the closure of A in its ambient space.
‣ Pullback( phi, A ) | ( operation ) |
Returns: an object in a thin category
Pullback A along the morphism defined by the ring homomorphism phi.
‣ EmbedInSmallerAmbientSpaceByEmbeddingAClosedSuperset( A ) | ( operation ) |
Returns: an object in a thin category
Embed A by embedding a closed superset of it in a smaller ambient space.
‣ EmbedInSmallerAmbientSpace( A ) | ( operation ) |
Returns: an object in a thin category
Embed A by embdeding its closure in a smaller ambient space.
gap> ZZ := HomalgRingOfIntegers( ); Z gap> A := ClosedSubsetOfSpec( "[ 10, 15 ]", ZZ ); V_{Z}( <...> ) gap> Display( A ); V( <10,15> ) gap> StandardizeObject( A ); V_{Z}( <...> ) gap> Display( A ); V( <5> ) gap> Ae := EmbedInSmallerAmbientSpace( A ); V_{GF(5)}( <...> ) gap> Display( Ae ); V( <> ) gap> I := ClosedSubsetOfSpec( "[ 1 ]", ZZ ); V_{Z}( <...> ) gap> Display( I ); V( <1> ) gap> Ie := EmbedInSmallerAmbientSpace( I ); V_{Z}( <...> ) gap> Display( Ie ); ∅ gap> Q := HomalgFieldOfRationals( ); Q gap> L := ClosedSubsetOfSpec( "[ 1 ]", Q ); V_{Q}( <...> ) gap> Display( L ); V( <1> ) gap> Le := EmbedInSmallerAmbientSpace( L ); V_{Q}( <...> ) gap> Display( Le ); ∅ gap> R := HomalgRingOfIntegersInSingular( ) * "x"; Z[x] gap> B := ClosedSubsetOfSpec( "[ 2, x^2+1 ]", R ); V_{Z[x]}( <...> ) gap> Display( B ); V( <2,x^2+1> ) gap> Be := EmbedInSmallerAmbientSpace( B ); V_{GF(2)}( <...> ) gap> Display( Be ); V( <> ) gap> C := ClosedSubsetOfSpec( "[ 2, x^2+x+1 ]", R ); V_{Z[x]}( <...> ) gap> Display( C ); V( <2,x^2+x+1> ) gap> Ce := EmbedInSmallerAmbientSpace( C ); V_{GF(2)[x]}( <...> ) gap> Display( Ce ); V( <x^2+x+1> )
‣ AClosedSingleton( A ) | ( operation ) |
Returns: an object in a Zariski coframe
If IsInitial( A ) = true an error is raised. Otherwise a subset consisting of single closed point of A is returned.
gap> Q := HomalgFieldOfRationals( ); Q gap> A := ClosedSubsetOfSpec( "", Q ); V_{Q}( <...> ) gap> p := AClosedSingleton( A ); V_{Q}( <...> ) gap> A = p; true
gap> Q := HomalgFieldOfRationals( ); Q gap> A := OpenSubsetOfSpec( "[1]", Q ); D_{Q}( <...> ) gap> p := AClosedSingleton( A ); V_{Q}( <...> ) gap> A = p; true
‣ PseudoIteratorOfClosedSingletons( A ) | ( operation ) |
Returns: an iterator
Returns a pseudo-iterator (without repetition) of closed singletons of A.
gap> F2 := HomalgRingOfIntegersInSingular( 2 ); GF(2) gap> F2t := F2 * "t"; GF(2)[t] gap> A := ClosedSubsetOfSpec( "[ 0 ]", F2t ); V_{GF(2)[t]}( <...> ) gap> iter := PseudoIteratorOfClosedSingletons( A ); <iterator of closed singletons of V_{GF(2)[t]}( <...> )> gap> iter2 := ShallowCopy( iter ); <iterator of closed singletons of V_{GF(2)[t]}( <...> )> gap> p := NextIterator( iter ); V_{GF(2)[t]}( <...> ) gap> Degree( p ); 1 gap> Display( p ); V( <t> ) gap> iter; <iterator of closed singletons of V_{GF(2)[t]}( I ) \ V_{GF(2)[t]}( J )> gap> iter2; <iterator of closed singletons of V_{GF(2)[t]}( <...> )> gap> p := NextIterator( iter ); V_{GF(2)[t]}( <...> ) gap> Degree( p ); 1 gap> Display( p ); V( <t+1> ) gap> p := NextIterator( iter ); V_{GF(2)[t]}( <...> ) gap> Degree( p ); 2 gap> Display( p ); V( <t^2+t+1> ) gap> p := NextIterator( iter ); V_{GF(2)[t]}( <...> ) gap> Degree( p ); 4 gap> Display( p ); V( <t^4+t+1> ) gap> p := NextIterator( iter );; gap> Degree( p ); 4 gap> Display( p ); V( <t^4+t^3+1> ) gap> p := NextIterator( iter );; gap> Degree( p ); 5 gap> Display( p ); V( <t^5+t^4+t^2+t+1> ) gap> p := NextIterator( iter );; gap> Degree( p ); 3 gap> Display( p ); V( <t^3+t+1> ) gap> p := NextIterator( iter );; gap> Degree( p ); 3 gap> Display( p ); V( <t^3+t^2+1> ) gap> p := NextIterator( iter );; gap> Degree( p ); 7 gap> Display( p ); V( <t^7+t^6+t^4+t^2+1> ) gap> p := NextIterator( iter );; gap> Degree( p ); 7 gap> Display( p ); V( <t^7+t^5+t^4+t^3+t^2+t+1> ) gap> List( [ 1 .. 20 ], a -> Degree( NextIterator( iter2 ) ) ); [ 1, 1, 2, 4, 4, 5, 3, 3, 7, 7, 7, 8, 8, 60, 4, 5, 7, 13, 20, 5 ] gap> List( [ 1 .. 10 ], a -> Degree( NextIterator( iter2 ) ) ); [ 43, 5, 6, 5, 8, 9, 73, 12, 8, 14 ]
‣ RingEpimorphismOfAClosedPoint( A ) | ( operation ) |
Returns: a homalg ring map
Return the ring epimorphism from the coordinate ring of the closure of A in its ambient space onto the residue field of a closed point of A.
‣ AClosedPoint( A ) | ( operation ) |
Returns: a homalg matrix
If IsInitial( A ) = true an error is raised. Otherwise a single closed point of A is returned.
‣ PseudoIteratorOfClosedPoints( A ) | ( operation ) |
Returns: an iterator
Returns a pseudo-iterator (without repetition) of closed points of A.
gap> ZZ := HomalgRingOfIntegersInSingular( ); Z gap> T := ClosedSubsetOfSpec( "", ZZ ); V_{Z}( <...> ) gap> Display( T ); V( <> ) gap> iter := PseudoIteratorOfClosedPoints( T ); <iterator of closed points of V_{Z}( <...> )> gap> iter2 := ShallowCopy( iter ); <iterator of closed points of V_{Z}( <...> )> gap> p := NextIterator( iter ); <An unevaluated 0 x 1 zero matrix over an external ring> gap> HomalgRing( p ); GF(2) gap> p := NextIterator( iter );; HomalgRing( p ); GF(3) gap> iter; <iterator of closed points of V_{Z}( I ) \ V_{Z}( J )> gap> List( [ 1 .. 10 ], i -> HomalgRing( NextIterator( iter2 ) ) ); [ GF(2), GF(3), GF(5), GF(7), GF(11), GF(13), GF(17), GF(19), GF(23), GF(29) ] gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q * "t"; Q[t] gap> V := ClosedSubsetOfSpec( "t^2-t", R ); V_{Q[t]}( <...> ) gap> itr := PseudoIteratorOfClosedPoints( V ); <iterator of closed points of V_{Q[t]}( <...> )> gap> IsDoneIterator( itr ); false gap> a := NextIterator( itr ); <An unevaluated 1 x 1 zero matrix over an external ring> gap> Display( a ); 0 gap> HomalgRing( a ); Q gap> b := NextIterator( itr ); <A 1 x 1 matrix over an external ring> gap> Display( b ); 1 gap> HomalgRing( b ); Q gap> IsDoneIterator( itr ); true
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