Ambient spaces¶

class sage.schemes.generic.ambient_space.AmbientSpace(n, R=Integer Ring)[source]¶

Bases: Scheme

Base class for ambient spaces over a ring.

INPUT:

n – dimension
R – ring

ambient_space()[source]¶

Return the ambient space of the scheme self, in this case self itself.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> P = ProjectiveSpace(Integer(4), ZZ)
>>> P.ambient_space() is P
True

>>> A = AffineSpace(Integer(2), GF(Integer(3)))
>>> A.ambient_space()
Affine Space of dimension 2 over Finite Field of size 3

base_extend(R)[source]¶

Return the natural extension of self over R.

INPUT:

R – a commutative ring, such that there is a natural map from the base ring of self to R

OUTPUT: an ambient space over R of the same structure as self

Note

A ValueError is raised if there is no such natural map. If you need to drop this condition, use self.change_ring(R).

EXAMPLES:

Sage

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))
Traceback (most recent call last):
...
ValueError: no natural map from the base ring (=Rational Field)
to R (=Finite Field of size 5)!

Python

>>> from sage.all import *
>>> P = ProjectiveSpace(Integer(2), ZZ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> PQ = P.base_extend(QQ); PQ
Projective Space of dimension 2 over Rational Field
>>> PQ.base_extend(GF(Integer(5)))
Traceback (most recent call last):
...
ValueError: no natural map from the base ring (=Rational Field)
to R (=Finite Field of size 5)!

change_ring(R)[source]¶

Return an ambient space over ring \(R\) and otherwise the same as self.

INPUT:

R – commutative ring

OUTPUT: ambient space over R

Note

There is no need to have any relation between \(R\) and the base ring of self, if you want to have such a relation, use self.base_extend(R) instead.

defining_polynomials()[source]¶

Return the defining polynomials of the scheme self. Since self is an ambient space, this is an empty list.

EXAMPLES:

Sage

sage: ProjectiveSpace(2, QQ).defining_polynomials()
()
sage: AffineSpace(0, ZZ).defining_polynomials()
()

Python

>>> from sage.all import *
>>> ProjectiveSpace(Integer(2), QQ).defining_polynomials()
()
>>> AffineSpace(Integer(0), ZZ).defining_polynomials()
()

dimension()[source]¶

Return the absolute dimension of this scheme.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> A2Q = AffineSpace(Integer(2), QQ)
>>> A2Q.dimension_absolute()
2
>>> A2Q.dimension()
2
>>> A2Z = AffineSpace(Integer(2), ZZ)
>>> A2Z.dimension_absolute()
3
>>> A2Z.dimension()
3

dimension_absolute()[source]¶

Return the absolute dimension of this scheme.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> A2Q = AffineSpace(Integer(2), QQ)
>>> A2Q.dimension_absolute()
2
>>> A2Q.dimension()
2
>>> A2Z = AffineSpace(Integer(2), ZZ)
>>> A2Z.dimension_absolute()
3
>>> A2Z.dimension()
3

dimension_relative()[source]¶

Return the relative dimension of this scheme over its base.

EXAMPLES:

Sage

sage: A2Q = AffineSpace(2, QQ)
sage: A2Q.dimension_relative()
2
sage: A2Z = AffineSpace(2, ZZ)
sage: A2Z.dimension_relative()
2

Python

>>> from sage.all import *
>>> A2Q = AffineSpace(Integer(2), QQ)
>>> A2Q.dimension_relative()
2
>>> A2Z = AffineSpace(Integer(2), ZZ)
>>> A2Z.dimension_relative()
2

gen(n=0)[source]¶

Return the \(n\)-th generator of the coordinate ring of the scheme self.

EXAMPLES:

Sage

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: P.gen(1)
y

Python

>>> from sage.all import *
>>> P = ProjectiveSpace(Integer(2), ZZ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> P.gen(Integer(1))
y

gens()[source]¶

Return the generators of the coordinate ring of the scheme self.

EXAMPLES:

Sage

sage: AffineSpace(0, QQ).gens()
()

sage: P.<x, y, z> = ProjectiveSpace(2, GF(5))
sage: P.gens()
(x, y, z)

Python

>>> from sage.all import *
>>> AffineSpace(Integer(0), QQ).gens()
()

>>> P = ProjectiveSpace(Integer(2), GF(Integer(5)), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> P.gens()
(x, y, z)

identity_morphism()[source]¶

Return the identity morphism.

OUTPUT: the identity morphism of the scheme self

EXAMPLES:

Sage

sage: A = AffineSpace(2, GF(3))
sage: A.identity_morphism()
Scheme endomorphism of Affine Space of dimension 2 over Finite Field of size 3
  Defn: Identity map

sage: P = ProjectiveSpace(3, ZZ)
sage: P.identity_morphism()
Scheme endomorphism of Projective Space of dimension 3 over Integer Ring
  Defn: Identity map

Python

>>> from sage.all import *
>>> A = AffineSpace(Integer(2), GF(Integer(3)))
>>> A.identity_morphism()
Scheme endomorphism of Affine Space of dimension 2 over Finite Field of size 3
  Defn: Identity map

>>> P = ProjectiveSpace(Integer(3), ZZ)
>>> P.identity_morphism()
Scheme endomorphism of Projective Space of dimension 3 over Integer Ring
  Defn: Identity map

is_projective()[source]¶

Return whether this ambient space is projective n-space.

EXAMPLES:

Sage

sage: AffineSpace(3, QQ).is_projective()
False
sage: ProjectiveSpace(3, QQ).is_projective()
True

Python

>>> from sage.all import *
>>> AffineSpace(Integer(3), QQ).is_projective()
False
>>> ProjectiveSpace(Integer(3), QQ).is_projective()
True

ngens()[source]¶

Return the number of generators of the coordinate ring of the scheme self.

EXAMPLES:

Sage

sage: AffineSpace(0, QQ).ngens()
0

sage: ProjectiveSpace(50, ZZ).ngens()
51

Python

>>> from sage.all import *
>>> AffineSpace(Integer(0), QQ).ngens()
0

>>> ProjectiveSpace(Integer(50), ZZ).ngens()
51

sage.schemes.generic.ambient_space.is_AmbientSpace(x)[source]¶

Return True if \(x\) is an ambient space.

EXAMPLES:

Sage

sage: from sage.schemes.generic.ambient_space import is_AmbientSpace
sage: is_AmbientSpace(ProjectiveSpace(3, ZZ))
doctest:warning...
DeprecationWarning: The function is_AmbientSpace is deprecated; use 'isinstance(..., AmbientSpace)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
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

Python

>>> from sage.all import *
>>> from sage.schemes.generic.ambient_space import is_AmbientSpace
>>> is_AmbientSpace(ProjectiveSpace(Integer(3), ZZ))
doctest:warning...
DeprecationWarning: The function is_AmbientSpace is deprecated; use 'isinstance(..., AmbientSpace)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
True
>>> is_AmbientSpace(AffineSpace(Integer(2), QQ))
True
>>> P = ProjectiveSpace(Integer(2), ZZ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> is_AmbientSpace(P.subscheme([x + y + z]))
False