For schemes and
, this module implements the set of morphisms
. This is done by SchemeHomset_generic.
As a special case, the Hom-sets can also represent the points of a
scheme. Recall that the -rational points of a scheme
over
can be identified with the set of morphisms
. In Sage
the rational points are implemented by such scheme morphisms. This is
done by SchemeHomset_points and its subclasses.
Note
You should not create the Hom-sets manually. Instead, use the Hom() method that is inherited by all schemes.
AUTHORS:
Bases: sage.structure.factory.UniqueFactory
Factory for Hom-sets of schemes.
EXAMPLES:
sage: A2 = AffineSpace(QQ,2)
sage: A3 = AffineSpace(QQ,3)
sage: Hom = A3.Hom(A2)
The Hom-sets are unique:
sage: Hom is copy(Hom)
True
sage: Hom is A3.Hom(A2)
True
sage: loads(Hom.dumps()) is Hom
True
Here is a tricky point. The Hom-sets are not identical if domains/codomains are isomorphic but not identiacal:
sage: A3_iso = AffineSpace(QQ,3)
sage: [ A3_iso is A3, A3_iso == A3 ]
[False, True]
sage: Hom_iso = A3_iso.Hom(A2)
sage: Hom_iso is Hom
False
sage: Hom_iso == Hom
True
TESTS:
sage: Hom.base()
Integer Ring
sage: Hom.base_ring()
Integer Ring
Create a key that uniquely determines the Hom-set.
INPUT:
EXAMPLES:
sage: A2 = AffineSpace(QQ,2)
sage: A3 = AffineSpace(QQ,3)
sage: A3.Hom(A2) # indirect doctest
Set of morphisms
From: Affine Space of dimension 3 over Rational Field
To: Affine Space of dimension 2 over Rational Field
sage: from sage.schemes.generic.homset import SchemeHomsetFactory
sage: SHOMfactory = SchemeHomsetFactory('test')
sage: key, extra = SHOMfactory.create_key_and_extra_args(A3,A2,check=False)
sage: key
(..., ..., Category of schemes over Integer Ring)
sage: extra
{'Y': Affine Space of dimension 2 over Rational Field,
'X': Affine Space of dimension 3 over Rational Field,
'base_ring': Integer Ring, 'check': False}
Create a SchemeHomset_generic.
INPUT:
EXAMPLES:
sage: A2 = AffineSpace(QQ,2)
sage: A3 = AffineSpace(QQ,3)
sage: A3.Hom(A2) is A3.Hom(A2) # indirect doctest
True
sage: from sage.schemes.generic.homset import SchemeHomsetFactory
sage: SHOMfactory = SchemeHomsetFactory('test')
sage: SHOMfactory.create_object(0, [id(A3),id(A2),A3.category()], check=True,
... X=A3, Y=A2, base_ring=QQ)
Set of morphisms
From: Affine Space of dimension 3 over Rational Field
To: Affine Space of dimension 2 over Rational Field
Bases: sage.categories.homset.HomsetWithBase
The base class for Hom-sets of schemes.
INPUT:
EXAMPLES:
sage: from sage.schemes.generic.homset import SchemeHomset_generic
sage: A2 = AffineSpace(QQ,2)
sage: Hom = SchemeHomset_generic(A2, A2); Hom
Set of morphisms
From: Affine Space of dimension 2 over Rational Field
To: Affine Space of dimension 2 over Rational Field
sage: Hom.category()
Category of hom sets in Category of Schemes
alias of SchemeMorphism
Return a natural map in the Hom space.
OUTPUT:
A SchemeMorphism if there is a natural map from domain to codomain. Otherwise, a NotImplementedError is raised.
EXAMPLES:
sage: A = AffineSpace(4, QQ)
sage: A.structure_morphism() # indirect doctest
Scheme morphism:
From: Affine Space of dimension 4 over Rational Field
To: Spectrum of Rational Field
Defn: Structure map
Bases: sage.schemes.generic.homset.SchemeHomset_generic
Set of rational points of the scheme.
Recall that the -rational points of a scheme
over
can
be identified with the set of morphisms
. In Sage,
the rational points are implemented by such scheme morphisms.
INPUT:
See SchemeHomset_generic.
EXAMPLES:
sage: from sage.schemes.generic.homset import SchemeHomset_points
sage: SchemeHomset_points(Spec(QQ), AffineSpace(ZZ,2))
Set of rational points of Affine Space of dimension 2 over Rational Field
Return the codomain with extended base, if necessary.
OUTPUT:
The codomain scheme, with its base ring extended to the
codomain. That is, the codomain is of the form and
the base ring of the domain is extended to
.
EXAMPLES:
sage: P2 = ProjectiveSpace(QQ,2)
sage: K.<a> = NumberField(x^2 + x - (3^3-3))
sage: K_points = P2(K); K_points
Set of rational points of Projective Space of dimension 2
over Number Field in a with defining polynomial x^2 + x - 24
sage: K_points.codomain()
Projective Space of dimension 2 over Rational Field
sage: K_points.extended_codomain()
Projective Space of dimension 2 over Number Field in a with
defining polynomial x^2 + x - 24
Return for a point Hom-set
.
OUTPUT:
A commutative ring.
EXAMPLES:
sage: P2 = ProjectiveSpace(ZZ,2)
sage: P2(QQ).value_ring()
Rational Field
Bases: sage.schemes.generic.homset.SchemeHomset_points_projective_field
Set of rational points of an abelian variety.
INPUT:
See SchemeHomset_generic.
TESTS:
The bug reported at trac #1785 is fixed:
sage: K.<a> = NumberField(x^2 + x - (3^3-3))
sage: E = EllipticCurve('37a')
sage: X = E(K)
sage: X
Abelian group of points on Elliptic Curve defined by
y^2 + y = x^3 + (-1)*x over Number Field in a with
defining polynomial x^2 + x - 24
sage: P = X([3,a])
sage: P
(3 : a : 1)
sage: P in E
False
sage: P in E.base_extend(K)
True
sage: P in X.codomain()
False
sage: P in X.extended_codomain()
True
Extend the base ring.
This is currently not implemented except for the trivial case R==ZZ.
INPUT:
EXAMPLES:
sage: E = EllipticCurve('37a')
sage: Hom = E.point_homset(); Hom
Abelian group of points on Elliptic Curve defined
by y^2 + y = x^3 - x over Rational Field
sage: Hom.base_ring()
Integer Ring
sage: Hom.base_extend(QQ)
Traceback (most recent call last):
...
NotImplementedError: Abelian variety point sets are not
implemented as modules over rings other than ZZ.
Bases: sage.schemes.generic.homset.SchemeHomset_points
Set of rational points of an affine variety.
INPUT:
See SchemeHomset_generic.
EXAMPLES:
sage: from sage.schemes.generic.homset import SchemeHomset_points_affine
sage: SchemeHomset_points_affine(Spec(QQ), AffineSpace(ZZ,2))
Set of rational points of Affine Space of dimension 2 over Rational Field
Return some or all rational points of an affine scheme.
INPUT:
OUTPUT:
EXAMPLES: The bug reported at #11526 is fixed:
sage: A2 = AffineSpace(ZZ,2)
sage: F = GF(3)
sage: A2(F).points()
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
sage: R = ZZ
sage: A.<x,y> = R[]
sage: I = A.ideal(x^2-y^2-1)
sage: V = AffineSpace(R,2)
sage: X = V.subscheme(I)
sage: M = X(R)
sage: M.points(1)
[(-1, 0), (1, 0)]
Bases: sage.schemes.generic.homset.SchemeHomset_points
Set of rational points of a projective variety over a field.
INPUT:
See SchemeHomset_generic.
EXAMPLES:
sage: from sage.schemes.generic.homset import SchemeHomset_points_projective_field
sage: SchemeHomset_points_projective_field(Spec(QQ), ProjectiveSpace(QQ,2))
Set of rational points of Projective Space of dimension 2 over Rational Field
Return some or all rational points of a projective scheme.
INPUT:
OUTPUT:
A list of points. Over a finite field, all points are returned. Over an infinite field, all points satisfying the bound are returned.
EXAMPLES:
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
Bases: sage.schemes.generic.homset.SchemeHomset_points
Set of rational points of a projective variety over a commutative ring.
INPUT:
See SchemeHomset_generic.
EXAMPLES:
sage: from sage.schemes.generic.homset import SchemeHomset_points_projective_ring
sage: SchemeHomset_points_projective_ring(Spec(ZZ), ProjectiveSpace(ZZ,2))
Set of rational points of Projective Space of dimension 2 over Integer Ring
Return some or all rational points of a projective scheme.
INPUT:
EXAMPLES:
sage: from sage.schemes.generic.homset import SchemeHomset_points_projective_ring
sage: H = SchemeHomset_points_projective_ring(Spec(ZZ), ProjectiveSpace(ZZ,2))
sage: H.points(3)
[(0 : 0 : 1), (0 : 1 : -3), (0 : 1 : -2), (0 : 1 : -1), (0 : 1 : 0), (0
: 1 : 1), (0 : 1 : 2), (0 : 1 : 3), (0 : 2 : -3), (0 : 2 : -1), (0 : 2 :
1), (0 : 2 : 3), (0 : 3 : -2), (0 : 3 : -1), (0 : 3 : 1), (0 : 3 : 2),
(1 : -3 : -3), (1 : -3 : -2), (1 : -3 : -1), (1 : -3 : 0), (1 : -3 : 1),
(1 : -3 : 2), (1 : -3 : 3), (1 : -2 : -3), (1 : -2 : -2), (1 : -2 : -1),
(1 : -2 : 0), (1 : -2 : 1), (1 : -2 : 2), (1 : -2 : 3), (1 : -1 : -3),
(1 : -1 : -2), (1 : -1 : -1), (1 : -1 : 0), (1 : -1 : 1), (1 : -1 : 2),
(1 : -1 : 3), (1 : 0 : -3), (1 : 0 : -2), (1 : 0 : -1), (1 : 0 : 0), (1
: 0 : 1), (1 : 0 : 2), (1 : 0 : 3), (1 : 1 : -3), (1 : 1 : -2), (1 : 1 :
-1), (1 : 1 : 0), (1 : 1 : 1), (1 : 1 : 2), (1 : 1 : 3), (1 : 2 : -3),
(1 : 2 : -2), (1 : 2 : -1), (1 : 2 : 0), (1 : 2 : 1), (1 : 2 : 2), (1 :
2 : 3), (1 : 3 : -3), (1 : 3 : -2), (1 : 3 : -1), (1 : 3 : 0), (1 : 3 :
1), (1 : 3 : 2), (1 : 3 : 3), (2 : -3 : -3), (2 : -3 : -2), (2 : -3 :
-1), (2 : -3 : 0), (2 : -3 : 1), (2 : -3 : 2), (2 : -3 : 3), (2 : -2 :
-3), (2 : -2 : -1), (2 : -2 : 1), (2 : -2 : 3), (2 : -1 : -3), (2 : -1 :
-2), (2 : -1 : -1), (2 : -1 : 0), (2 : -1 : 1), (2 : -1 : 2), (2 : -1 :
3), (2 : 0 : -3), (2 : 0 : -1), (2 : 0 : 1), (2 : 0 : 3), (2 : 1 : -3),
(2 : 1 : -2), (2 : 1 : -1), (2 : 1 : 0), (2 : 1 : 1), (2 : 1 : 2), (2 :
1 : 3), (2 : 2 : -3), (2 : 2 : -1), (2 : 2 : 1), (2 : 2 : 3), (2 : 3 :
-3), (2 : 3 : -2), (2 : 3 : -1), (2 : 3 : 0), (2 : 3 : 1), (2 : 3 : 2),
(2 : 3 : 3), (3 : -3 : -2), (3 : -3 : -1), (3 : -3 : 1), (3 : -3 : 2),
(3 : -2 : -3), (3 : -2 : -2), (3 : -2 : -1), (3 : -2 : 0), (3 : -2 : 1),
(3 : -2 : 2), (3 : -2 : 3), (3 : -1 : -3), (3 : -1 : -2), (3 : -1 : -1),
(3 : -1 : 0), (3 : -1 : 1), (3 : -1 : 2), (3 : -1 : 3), (3 : 0 : -2), (3
: 0 : -1), (3 : 0 : 1), (3 : 0 : 2), (3 : 1 : -3), (3 : 1 : -2), (3 : 1
: -1), (3 : 1 : 0), (3 : 1 : 1), (3 : 1 : 2), (3 : 1 : 3), (3 : 2 : -3),
(3 : 2 : -2), (3 : 2 : -1), (3 : 2 : 0), (3 : 2 : 1), (3 : 2 : 2), (3 :
2 : 3), (3 : 3 : -2), (3 : 3 : -1), (3 : 3 : 1), (3 : 3 : 2)]
Bases: sage.schemes.generic.homset.SchemeHomset_generic
Set of rational points of an affine variety.
INPUT:
See SchemeHomset_generic.
EXAMPLES:
sage: from sage.schemes.generic.homset import SchemeHomset_points_spec
sage: SchemeHomset_points_spec(Spec(QQ), Spec(QQ))
Set of rational points of Spectrum of Rational Field
Test whether H is a scheme Hom-set.
EXAMPLES:
sage: f = Spec(QQ).identity_morphism(); f
Scheme endomorphism of Spectrum of Rational Field
Defn: Identity map
sage: from sage.schemes.generic.homset import is_SchemeHomset
sage: is_SchemeHomset(f)
False
sage: is_SchemeHomset(f.parent())
True
sage: is_SchemeHomset('a string')
False