Set of homomorphisms between two schemes

For schemes \(X\) and \(Y\), this module implements the set of morphisms \(\mathrm{Hom}(X,Y)\). 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 \(\mathrm{Spec}(K) \to X\). 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[source]

Bases: UniqueFactory

Factory for Hom-sets of schemes.

EXAMPLES:

sage: A2 = AffineSpace(QQ, 2)
sage: A3 = AffineSpace(QQ, 3)
sage: Hom = A3.Hom(A2)

>>> from sage.all import *
>>> A2 = AffineSpace(QQ, Integer(2))
>>> A3 = AffineSpace(QQ, Integer(3))
>>> Hom = A3.Hom(A2)

The Hom-sets are uniquely determined by domain and codomain:

sage: Hom is copy(Hom)
True
sage: Hom is A3.Hom(A2)
True

>>> from sage.all import *
>>> Hom is copy(Hom)
True
>>> Hom is A3.Hom(A2)
True

The Hom-sets are identical if the domains and codomains are identical:

sage: loads(Hom.dumps()) is Hom
True
sage: A3_iso = AffineSpace(QQ, 3)
sage: A3_iso is A3
True
sage: Hom_iso = A3_iso.Hom(A2)
sage: Hom_iso is Hom
True

>>> from sage.all import *
>>> loads(Hom.dumps()) is Hom
True
>>> A3_iso = AffineSpace(QQ, Integer(3))
>>> A3_iso is A3
True
>>> Hom_iso = A3_iso.Hom(A2)
>>> Hom_iso is Hom
True
create_key_and_extra_args(X, Y, category=None, base=None, check=True, as_point_homset=False)[source]

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 Rational Field, False)
sage: extra
{'X': Affine Space of dimension 3 over Rational Field,
 'Y': Affine Space of dimension 2 over Rational Field,
 'base_ring': Rational Field,
 'check': False}

>>> from sage.all import *
>>> A2 = AffineSpace(QQ, Integer(2))
>>> A3 = AffineSpace(QQ, Integer(3))
>>> 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
>>> from sage.schemes.generic.homset import SchemeHomsetFactory
>>> SHOMfactory = SchemeHomsetFactory('test')
>>> key, extra = SHOMfactory.create_key_and_extra_args(A3, A2, check=False)
>>> key
(..., ..., Category of schemes over Rational Field, False)
>>> extra
{'X': Affine Space of dimension 3 over Rational Field,
 'Y': Affine Space of dimension 2 over Rational Field,
 'base_ring': Rational Field,
 'check': False}
create_object(version, key, **extra_args)[source]

Create a SchemeHomset_generic.

INPUT:

  • version – object version; currently not used

  • key – a key created by create_key_and_extra_args()

  • extra_args – 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(), False],
....:                           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

>>> from sage.all import *
>>> A2 = AffineSpace(QQ, Integer(2))
>>> A3 = AffineSpace(QQ, Integer(3))
>>> A3.Hom(A2) is A3.Hom(A2)   # indirect doctest
True
>>> from sage.schemes.generic.homset import SchemeHomsetFactory
>>> SHOMfactory = SchemeHomsetFactory('test')
>>> SHOMfactory.create_object(Integer(0), [id(A3), id(A2), A3.category(), False],
...                           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)[source]

Bases: 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 (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 endsets of schemes over Rational Field

>>> from sage.all import *
>>> from sage.schemes.generic.homset import SchemeHomset_generic
>>> A2 = AffineSpace(QQ,Integer(2))
>>> 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
>>> Hom.category()
Category of endsets of schemes over Rational Field
Element[source]

alias of SchemeMorphism

natural_map()[source]

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

>>> from sage.all import *
>>> A = AffineSpace(Integer(4), QQ)
>>> 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)[source]

Bases: 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 \(\mathrm{Spec}(K) \to X\). In Sage, the rational points are implemented by such scheme morphisms.

If a scheme has a finite number of points, then the homset is supposed to implement the Python iterator interface. See SchemeHomset_points_toric_field for example.

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

>>> from sage.all import *
>>> from sage.schemes.generic.homset import SchemeHomset_points
>>> SchemeHomset_points(Spec(QQ), AffineSpace(ZZ,Integer(2)))
Set of rational points of Affine Space of dimension 2 over Rational Field
cardinality()[source]

Return the number of points.

OUTPUT: integer or infinity

EXAMPLES:

sage: toric_varieties.P2().point_set().cardinality()                        # needs sage.geometry.polyhedron sage.graphs
+Infinity

sage: P2 = toric_varieties.P2(base_ring=GF(3))                              # needs sage.geometry.polyhedron sage.graphs
sage: P2.point_set().cardinality()                                          # needs sage.geometry.polyhedron sage.graphs
13

>>> from sage.all import *
>>> toric_varieties.P2().point_set().cardinality()                        # needs sage.geometry.polyhedron sage.graphs
+Infinity

>>> P2 = toric_varieties.P2(base_ring=GF(Integer(3)))                              # needs sage.geometry.polyhedron sage.graphs
>>> P2.point_set().cardinality()                                          # needs sage.geometry.polyhedron sage.graphs
13
extended_codomain()[source]

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

EXAMPLES:

sage: # needs sage.rings.number_field
sage: P2 = ProjectiveSpace(QQ, 2)
sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> P2 = ProjectiveSpace(QQ, Integer(2))
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + x - (Integer(3)**Integer(3)-Integer(3)), names=('a',)); (a,) = K._first_ngens(1)
>>> 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
>>> K_points.codomain()
Projective Space of dimension 2 over Rational Field
>>> K_points.extended_codomain()
Projective Space of dimension 2
 over Number Field in a with defining polynomial x^2 + x - 24
list()[source]

Return a tuple containing all points.

OUTPUT: a tuple containing all points of the toric variety

EXAMPLES:

sage: P1 = toric_varieties.P1(base_ring=GF(3))                              # needs sage.geometry.polyhedron sage.graphs
sage: P1.point_set().list()                                                 # needs sage.geometry.polyhedron sage.graphs
([0 : 1], [1 : 0], [1 : 1], [1 : 2])

>>> from sage.all import *
>>> P1 = toric_varieties.P1(base_ring=GF(Integer(3)))                              # needs sage.geometry.polyhedron sage.graphs
>>> P1.point_set().list()                                                 # needs sage.geometry.polyhedron sage.graphs
([0 : 1], [1 : 0], [1 : 1], [1 : 2])
value_ring()[source]

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

OUTPUT: a commutative ring

EXAMPLES:

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

>>> from sage.all import *
>>> P2 = ProjectiveSpace(ZZ, Integer(2))
>>> P2(QQ).value_ring()
Rational Field
zero()[source]

Return the identity of the codomain with extended base, if necessary.

sage.schemes.generic.homset.is_SchemeHomset(H)[source]

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)
doctest:warning...
DeprecationWarning: The function is_SchemeHomset is deprecated; use 'isinstance(..., SchemeHomset_generic)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
False
sage: is_SchemeHomset(f.parent())
True
sage: is_SchemeHomset('a string')
False

>>> from sage.all import *
>>> f = Spec(QQ).identity_morphism();  f
Scheme endomorphism of Spectrum of Rational Field
  Defn: Identity map
>>> from sage.schemes.generic.homset import is_SchemeHomset
>>> is_SchemeHomset(f)
doctest:warning...
DeprecationWarning: The function is_SchemeHomset is deprecated; use 'isinstance(..., SchemeHomset_generic)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
False
>>> is_SchemeHomset(f.parent())
True
>>> is_SchemeHomset('a string')
False