def _Hom_(self, Y, category=None, check=True):
"""
Return the set of scheme morphisms from ``self`` to ``Y``.
INPUT:
- ``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.
OUTPUT:
The set of morphisms from ``self`` to ``Y``.
EXAMPLES::
sage: P = ProjectiveSpace(ZZ, 3)
sage: S = Spec(ZZ)
sage: S._Hom_(P)
Set of rational points of Projective Space of dimension 3 over Integer Ring
"""
from sage.schemes.generic.homset import SchemeHomset
return SchemeHomset(self, Y, category=category, check=check)
def _Hom_(self, Y, category=None, check=True):
"""
Return the set of scheme morphisms from ``self`` to ``Y``.
INPUT:
- ``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.
OUTPUT:
The set of morphisms from ``self`` to ``Y``.
EXAMPLES::
sage: P = ProjectiveSpace(ZZ, 3)
sage: S = Spec(ZZ)
sage: S._Hom_(P)
Set of morphisms
From: Spectrum of Integer Ring
To: Projective Space of dimension 3 over Integer Ring
TESTS::
sage: S._Hom_(P).__class__
<class 'sage.schemes.generic.homset.SchemeHomset_generic_with_category'>
sage: E = EllipticCurve('37a1')
sage: Hom(E, E).__class__
<class 'sage.schemes.generic.homset.SchemeHomset_generic_with_category'>
sage: Hom(Spec(ZZ), Spec(ZZ)).__class__
<class 'sage.schemes.generic.homset.SchemeHomset_generic_with_category_with_equality_by_id'>
"""
from sage.schemes.generic.homset import SchemeHomset
return SchemeHomset(self, Y, category=category, check=check)
def _Hom_(self, Y, category=None, check=True):
"""
Return the set of scheme morphisms from ``self`` to ``Y``.
INPUT:
- ``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.
OUTPUT:
The set of morphisms from ``self`` to ``Y``.
EXAMPLES::
sage: P = ProjectiveSpace(ZZ, 3)
sage: S = Spec(ZZ)
sage: S._Hom_(P)
Set of rational points of Projective Space of dimension 3 over Integer Ring
TESTS::
sage: S._Hom_(P).__class__
<class 'sage.schemes.projective.projective_homset.SchemeHomset_points_projective_ring_with_category'>
sage: E = EllipticCurve('37a1')
sage: Hom(E, E).__class__
<class 'sage.schemes.generic.homset.SchemeHomset_generic_with_category'>
sage: Hom(Spec(ZZ), Spec(ZZ)).__class__
<class 'sage.schemes.affine.affine_homset.SchemeHomset_points_spec_with_category'>
"""
from sage.schemes.generic.homset import SchemeHomset
return SchemeHomset(self, Y, category=category, check=check)
def point_homset(self, S=None):
"""
Return the set of S-valued points of this scheme.
INPUT:
- ``S`` -- a commutative ring.
OUTPUT:
The set of morphisms `Spec(S)\to X`.
EXAMPLES::
sage: P = ProjectiveSpace(ZZ, 3)
sage: P.point_homset(ZZ)
Set of rational points of Projective Space of dimension 3 over Integer Ring
sage: P.point_homset(QQ)
Set of rational points of Projective Space of dimension 3 over Rational Field
sage: P.point_homset(GF(11))
Set of rational points of Projective Space of dimension 3 over
Finite Field of size 11
TESTS::
sage: P = ProjectiveSpace(QQ,3)
sage: P.point_homset(GF(11))
Traceback (most recent call last):
...
ValueError: There must be a natural map S --> R, but
S = Rational Field and R = Finite Field of size 11
"""
if S is None:
S = self.base_ring()
from sage.schemes.generic.spec import Spec
SpecS = Spec(S, self.base_ring())
from sage.schemes.generic.homset import SchemeHomset
return SchemeHomset(SpecS, self)
def __new__(cls, R, S, category):
"""
TESTS::
sage: E = EllipticCurve('37a1')
sage: Hom(E, E).__class__
<class 'sage.schemes.generic.homset.SchemeHomset_generic_with_category'>
If both schemes R and S are actually specs, we want
the parent for Hom(R, S) to be in a different class::
sage: Hom(Spec(ZZ), Spec(ZZ)).__class__
<class 'sage.schemes.generic.homset.SchemeHomset_points_spec_with_category'>
Currently, and to minimize the changes, this is done
by delegating the job to SchemeHomset. This is not
very robust: for example, only one category can do
this hack.
FIXME: this might be better handled by an extra Spec category
"""
from sage.schemes.generic.homset import SchemeHomset
return SchemeHomset(R, S, category=category)