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1 (Co)frames/Locales of Zariski closed/open subsets
 1.1 Info Class
 1.2 GAP Categories
 1.3 Properties
 1.4 Attributes
 1.5 Operations

1 (Co)frames/Locales of Zariski closed/open subsets

1.1 Info Class

1.1-1 InfoZariskiFrames
‣ InfoZariskiFrames( info class )

1.2 GAP Categories

1.2-1 IsZariskiFrameOrCoframe
‣ IsZariskiFrameOrCoframe( object )( filter )

Returns: true or false

The GAP category of objects in a Zariski frame or coframe.

1.2-2 IsObjectInZariskiFrameOrCoframe
‣ IsObjectInZariskiFrameOrCoframe( object )( filter )

Returns: true or false

The GAP category of objects in a Zariski frame or coframe.

1.2-3 IsMorphismInZariskiFrameOrCoframe
‣ IsMorphismInZariskiFrameOrCoframe( morphism )( filter )

Returns: true or false

The GAP category of morphisms in a Zariski frame or coframe.

1.3 Properties

1.3-1 IsOpen
‣ IsOpen( arg )( property )

Returns: true or false

1.3-2 IsClosed
‣ IsClosed( arg )( operation )

1.4 Attributes

1.4-1 UnderlyingRing
‣ UnderlyingRing( arg )( attribute )

1.4-2 BaseOfFibration
‣ BaseOfFibration( arg )( attribute )

1.4-3 BaseOfFibration
‣ BaseOfFibration( arg )( attribute )

1.4-4 Closure
‣ Closure( A )( attribute )

Returns: an object in the Zariski coframe

The closure of A.

1.4-5 ParametrizedObject
‣ ParametrizedObject( A )( attribute )

1.4-6 ParametrizedObject
‣ ParametrizedObject( A )( attribute )

1.4-7 ParametrizedObject
‣ ParametrizedObject( A )( attribute )

1.4-8 LocallyClosedPart
‣ LocallyClosedPart( A )( attribute )

1.4-9 DistinguishedLocallyClosedPart
‣ DistinguishedLocallyClosedPart( A )( attribute )

1.4-10 DistinguishedLocallyClosedPart
‣ DistinguishedLocallyClosedPart( A )( attribute )

1.4-11 DistinguishedLocallyClosedPart
‣ DistinguishedLocallyClosedPart( A )( attribute )

1.4-12 AffineApproximation
‣ AffineApproximation( A )( attribute )

1.4-13 CanonicalObject
‣ CanonicalObject( A )( attribute )

1.4-14 RabinowitschCover
‣ 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.

1.5 Operations

1.5-1 AClosedSuperset
‣ 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.

1.5-2 BestUnderlyingColumn
‣ BestUnderlyingColumn( A )( operation )

Returns: a homalg matrix

The column matrix of generators of the defining ideal of A.

1.5-3 NormalizeObject
‣ NormalizeObject( A )( operation )

Returns: the object in the Zariski frame or coframe A

1.5-4 StandardizeObject
‣ StandardizeObject( A )( operation )

Returns: the object in the Zariski frame or coframe A

1.5-5 IsSubset
‣ IsSubset( A, B )( operation )

Returns: true or false

Check if A is bigger than B w.r.t. inclusion.

1.5-6 RingEpimorphismOfClosedSuperset
‣ 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.

1.5-7 RingEpimorphismOfClosure
‣ 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.

1.5-8 Pullback
‣ Pullback( phi, A )( operation )

Returns: an object in a thin category

Pullback A along the morphism defined by the ring homomorphism phi.

1.5-9 EmbedInSmallerAmbientSpaceByEmbeddingAClosedSuperset
‣ EmbedInSmallerAmbientSpaceByEmbeddingAClosedSuperset( A )( operation )

Returns: an object in a thin category

Embed A by embedding a closed superset of it in a smaller ambient space.

1.5-10 EmbedInSmallerAmbientSpace
‣ 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> )

1.5-11 AClosedSingleton
‣ 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

1.5-12 PseudoIteratorOfClosedSingletons
‣ 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 ]

1.5-13 RingEpimorphismOfAClosedPoint
‣ 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.

1.5-14 AClosedPoint
‣ AClosedPoint( A )( operation )

Returns: a homalg matrix

If IsInitial( A ) = true an error is raised. Otherwise a single closed point of A is returned.

1.5-15 PseudoIteratorOfClosedPoints
‣ 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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