Set of homomorphisms between two schemes

For schemes \(X\) and \(Y\), this module implements the set of morphisms \(Hom(X,Y)\). This is done by SchemeHomset_generic.

As a special case, the Hom-sets can also represent the points of a scheme. Recall that the \(K\)-rational points of a scheme \(X\) over \(k\) can be identified with the set of morphisms \(Spec(K) \to X\). 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:

• William Stein (2006): initial version.
• Volker Braun (2011-08-11): significant improvement and refactoring.
• Ben Hutz (June 2012): added support for projective ring

class sage.schemes.generic.homset.SchemeHomsetFactory

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_key_and_extra_args(X, Y, category=None, base=Integer Ring, check=True)

Create a key that uniquely determines the Hom-set.

INPUT:

• X – a scheme. The domain of the morphisms.
• Y – a scheme. The codomain of the morphisms.
• category – a category for the Hom-sets (default: schemes over given base).
• base – a scheme or a ring. The base scheme of domain and codomain schemes. If a ring is specified, the spectrum of that ring will be used as base scheme.
• check – boolean (default: True).

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_object(version, key, **extra_args)

Create a SchemeHomset_generic.

INPUT:

• version – object version. Currently not used.
• key – a key created by create_key_and_extra_args().
• extra_args – a dictionary of extra keyword arguments.

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

class sage.schemes.generic.homset.SchemeHomset_generic(X, Y, category=None, check=True, base=None)

Bases: sage.categories.homset.HomsetWithBase

The base class for Hom-sets of schemes.

INPUT:

• X – a scheme. The domain of the Hom-set.
• Y – a scheme. The codomain of the Hom-set.
• category – a category (optional). The category of the Hom-set.
• check – boolean (optional, default=``True``). Whether to check the defining data for consistency.

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

Element

alias of SchemeMorphism

natural_map()

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

class sage.schemes.generic.homset.SchemeHomset_points(X, Y, category=None, check=True, base=Integer Ring)

Bases: sage.schemes.generic.homset.SchemeHomset_generic

Set of rational points of the scheme.

Recall that the \(K\)-rational points of a scheme \(X\) over \(k\) can be identified with the set of morphisms \(Spec(K) o X\). 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

extended_codomain()

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 \(Spec(R)\) and the base ring of the domain is extended to \(R\).

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

value_ring()

Return \(R\) for a point Hom-set \(X(Spec(R))\).

OUTPUT:

A commutative ring.

EXAMPLES:

sage: P2 = ProjectiveSpace(ZZ,2)
sage: P2(QQ).value_ring()
Rational Field

class sage.schemes.generic.homset.SchemeHomset_points_abelian_variety_field(X, Y, category=None, check=True, base=Integer Ring)

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

base_extend(R)

Extend the base ring.

This is currently not implemented except for the trivial case R==ZZ.

INPUT:

• R – a ring.

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.

class sage.schemes.generic.homset.SchemeHomset_points_affine(X, Y, category=None, check=True, base=Integer Ring)

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

points(B=0)

Return some or all rational points of an affine scheme.

INPUT:

• B – integer (optional, default: 0). The bound for the height of the coordinates.

OUTPUT:

• If the base ring is a finite field: all points of the scheme, given by coordinate tuples.
• If the base ring is \(\QQ\) or \(\ZZ\): the subset of points whose coordinates have height B or less.

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)]

class sage.schemes.generic.homset.SchemeHomset_points_projective_field(X, Y, category=None, check=True, base=Integer Ring)

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

points(B=0)

Return some or all rational points of a projective scheme.

INPUT:

• \(B\) – integer (optional, default=0). The bound for the coordinates.

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)]

class sage.schemes.generic.homset.SchemeHomset_points_projective_ring(X, Y, category=None, check=True, base=Integer Ring)

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

points(B=0)

Return some or all rational points of a projective scheme.

INPUT:

• \(B\) – integer (optional, default=0). The bound for the coordinates.

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)]

class sage.schemes.generic.homset.SchemeHomset_points_spec(X, Y, category=None, check=True, base=None)

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

sage.schemes.generic.homset.is_SchemeHomset(H)

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