Base class for ambient spaces over a ring.
INPUT:
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
sage: A.__cmp__(ProjectiveSpace(2, QQ))
...
NotImplementedError
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
Verify that the coordinates of v define a point on this scheme, or raise a TypeError.
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
sage: A._check_satisfies_equations([1, 2, 3, 4, 5])
...
NotImplementedError
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
sage: A._constructor()
...
NotImplementedError
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
sage: A._latex_()
...
NotImplementedError
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
sage: A._latex_generic_point([1, 2, 3, 4, 5])
...
NotImplementedError
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
sage: A._repr_()
...
NotImplementedError
TEST:
sage: from sage.schemes.generic.ambient_space import AmbientSpace
sage: A = AmbientSpace(5, ZZ)
sage: A._repr_generic_point([1, 2, 3, 4, 5])
...
NotImplementedError
Return the ambient space of the scheme self, in this case self itself.
EXAMPLES:
sage: P = ProjectiveSpace(4, ZZ)
sage: P.ambient_space() is P
True
sage: A = AffineSpace(2, GF(3))
sage: A.ambient_space()
Affine Space of dimension 2 over Finite Field of size 3
Return the base change of self to the ring , via the natural
map from the base ring of self to
.
A ValueError is raised if there is no natural map between the two rings.
EXAMPLES:
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: PQ = P.base_extend(QQ); PQ
Projective Space of dimension 2 over Rational Field
sage: PQ.base_extend(GF(5))
...
ValueError: No natural map from the base ring (=Rational Field) to S (=Finite Field of size 5)
Return the defining polynomials of the scheme self. Since self is an ambient space, this is an empty list.
EXAMPLES:
sage: ProjectiveSpace(2, QQ).defining_polynomials()
()
sage: AffineSpace(0, ZZ).defining_polynomials()
()
Return the absolute dimension of this scheme.
EXAMPLES:
sage: A2Q = AffineSpace(2, QQ)
sage: A2Q.dimension_absolute()
2
sage: A2Q.dimension()
2
sage: A2Z = AffineSpace(2, ZZ)
sage: A2Z.dimension_absolute()
3
sage: A2Z.dimension()
3
Return the absolute dimension of this scheme.
EXAMPLES:
sage: A2Q = AffineSpace(2, QQ)
sage: A2Q.dimension_absolute()
2
sage: A2Q.dimension()
2
sage: A2Z = AffineSpace(2, ZZ)
sage: A2Z.dimension_absolute()
3
sage: A2Z.dimension()
3
Return the relative dimension of this scheme over its base.
EXAMPLES:
sage: A2Q = AffineSpace(2, QQ)
sage: A2Q.dimension_relative()
2
sage: A2Z = AffineSpace(2, ZZ)
sage: A2Z.dimension_relative()
2
Return the -th generator of the coordinate ring of the
scheme self.
EXAMPLES:
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: P.gen(1)
y
Return the generators of the coordinate ring of the scheme self.
EXAMPLES:
sage: AffineSpace(0, QQ).gens()
()
sage: P.<x, y, z> = ProjectiveSpace(2, GF(5))
sage: P.gens()
(x, y, z)
Return the number of generators of the coordinate ring of the scheme self.
EXAMPLES:
sage: AffineSpace(0, QQ).ngens()
0
sage: ProjectiveSpace(50, ZZ).ngens()
51
Return True if is an ambient space.
EXAMPLES:
sage: from sage.schemes.generic.ambient_space import is_AmbientSpace
sage: is_AmbientSpace(ProjectiveSpace(3, ZZ))
True
sage: is_AmbientSpace(AffineSpace(2, QQ))
True
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: is_AmbientSpace(P.subscheme([x+y+z]))
False