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:
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 uniquely determined by domain and codomain:
sage: Hom is copy(Hom) True sage: 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
- create_key_and_extra_args(X, Y, category=None, base=None, check=True, as_point_homset=False)#
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}
- create_object(version, key, **extra_args)#
Create a
SchemeHomset_generic
.INPUT:
version
– object version. Currently not used.key
– a key created bycreate_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(), 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)#
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 (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 endsets of schemes over Rational Field
- 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, aNotImplementedError
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:
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.
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
- cardinality()#
Return the number of points.
OUTPUT:
An 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
- 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: # 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
- list()#
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])
- 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
- 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