Schemes

AUTHORS:

  • William Stein, David Kohel, Kiran Kedlaya (2008): added zeta_series
class sage.schemes.generic.scheme.AffineScheme(X)

An abstract affine scheme.

hom(x, Y=None)

Return the scheme morphism from self to Y defined by x.

If Y is not given, try to determine from context.

EXAMPLES: We construct the inclusion from \mathrm{Spec}(\QQ) into \mathrm{Spec}(\ZZ) induced by the inclusion from \ZZ into \QQ.

sage: X = Spec(QQ)
sage: X.hom(ZZ.hom(QQ))
Affine Scheme morphism:
  From: Spectrum of Rational Field
  To:   Spectrum of Integer Ring
  Defn: Ring Coercion morphism:
          From: Integer Ring
          To:   Rational Field
class sage.schemes.generic.scheme.Scheme(X)
_Hom_(Y, cat=None, check=True)

Return the set of scheme morphisms from self to Y.

EXAMPLES:

sage: P = ProjectiveSpace(ZZ, 3)
sage: S = Spec(ZZ)
sage: S._Hom_(P)
Set of points of Projective Space of dimension 3 over Integer Ring defined over Integer Ring
__add__(X)

Return the disjoint union of the schemes self and X.

EXAMPLES:

sage: S = Spec(QQ)
sage: X = AffineSpace(1, QQ)
sage: S.union(X)
...
NotImplementedError
__call__(*args)

If S is a ring or scheme, return the set X(S) of S-valued points on X. If S is a list or tuple or just the coordinates, return a point in X(T), where T is the base scheme of self.

EXAMPLES:

sage: A = AffineSpace(2, QQ)

We create some point sets:

sage: A(QQ)
Set of Rational Points of Affine Space of dimension 2 over Rational Field            
sage: A(RR)
Set of Rational Points of Affine Space of dimension 2 over Real Field with 53 bits of precision

Space of dimension 2 over Rational Field

sage: R.<x> = PolynomialRing(QQ)
sage: A(NumberField(x^2+1, 'a'))
Set of Rational Points of Affine Space of dimension 2 over Number Field in a with defining polynomial x^2 + 1
sage: A(GF(7))
...
ValueError: No natural map from the base ring (=Rational Field) to S (=Finite Field of size 7)

We create some points:

sage: A(QQ)([1,0])
(1, 0)

We create the same point by giving the coordinates of the point directly.

sage: A( 1,0 )
(1, 0)
__cmp__(X)

EXAMPLES:

sage: X = Spec(QQ);  Y = Spec(QQ)
sage: X == Y
True
sage: X is Y
False
__div__(Y)

Return the base extension of self to Y.

EXAMPLES:

sage: A = AffineSpace(3, ZZ)
sage: A
Affine Space of dimension 3 over Integer Ring
sage: A/QQ
Affine Space of dimension 3 over Rational Field
sage: A/GF(7)
Affine Space of dimension 3 over Finite Field of size 7
__init__(X)

A scheme.

TESTS:

sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: RmodI = R.quotient(I)
sage: X = Spec(RmodI)
sage: X == loads(dumps(X))
True
_homset_class()

EXAMPLES:

sage: X = Spec(QQ)
sage: X._homset_class()
...
NotImplementedError
_point_class()

EXAMPLES:

sage: X = Spec(QQ)
sage: X._point_class()
...
NotImplementedError
_point_morphism_class()

EXAMPLES:

sage: X = Spec(QQ)
sage: X._point_morphism_class()
...
NotImplementedError
base_extend(Y)

Y is either a scheme in the same category as self or a ring.

EXAMPLES:

sage: X = Spec(QQ)
sage: X.base_scheme()
Spectrum of Integer Ring
sage: X.base_extend(QQ)
...
NotImplementedError
base_morphism()

Return the structure morphism from the scheme self to its base scheme.

EXAMPLES:

sage: A = AffineSpace(4, QQ)
sage: A.base_morphism()
Scheme morphism:
  From: Affine Space of dimension 4 over Rational Field
  To:   Spectrum of Rational Field
  Defn: Structure map
sage: X = Spec(QQ)
sage: X.base_morphism()
Scheme morphism:
  From: Spectrum of Rational Field
  To:   Spectrum of Integer Ring
  Defn: Structure map
base_ring()

Return the base ring of the scheme self.

EXAMPLES:

sage: A = AffineSpace(4, QQ)
sage: A.base_ring()
Rational Field
sage: X = Spec(QQ)
sage: X.base_ring()
Integer Ring
base_scheme()

Return the base scheme of the scheme self.

EXAMPLES:

sage: A = AffineSpace(4, QQ)
sage: A.base_scheme()
Spectrum of Rational Field
sage: X = Spec(QQ)
sage: X.base_scheme()
Spectrum of Integer Ring
category()

Return the category to which this scheme belongs. This is the category of all schemes over the base scheme of self.

EXAMPLES:

sage: ProjectiveSpace(4, QQ).category()
Category of schemes over Spectrum of Rational Field
coordinate_ring()

Return the coordinate ring of this scheme, if defined. Otherwise raise a ValueError.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: X = Spec(R.quotient(I))
sage: X.coordinate_ring()
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 - y^2)
count_points(n)

Count points over \GF{q}, \ldots, \GF{q^n} on a scheme over a finite field \GF{q}.

Note

This is currently only implemented for schemes over prime order finite fields.

EXAMPLES:

sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: C.count_points(4)
[6, 12, 18, 96]
sage: C.base_extend(GF(9,'a')).count_points(2)
...
NotImplementedError: Point counting only implemented for schemes over prime fields
dimension()

Return the absolute dimension of this scheme.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: X = Spec(R.quotient(I))
sage: X.dimension_absolute()
...
NotImplementedError
sage: X.dimension()
...
NotImplementedError
dimension_absolute()

Return the absolute dimension of this scheme.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: X = Spec(R.quotient(I))
sage: X.dimension_absolute()
...
NotImplementedError
sage: X.dimension()
...
NotImplementedError
dimension_relative()

Return the relative dimension of this scheme over its base.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: X = Spec(R.quotient(I))
sage: X.dimension_relative()
...
NotImplementedError
hom(x, Y=None)

Return the scheme morphism from self to Y defined by x. If x is a scheme, try to determine a natural map to x.

If Y is not given, try to determine Y from context.

EXAMPLES:

sage: P = ProjectiveSpace(ZZ, 3)
sage: P.hom(Spec(ZZ))
Scheme morphism:
  From: Projective Space of dimension 3 over Integer Ring
  To:   Spectrum of Integer Ring
  Defn: Structure map
identity_morphism()

Return the identity morphism of the scheme self.

EXAMPLES:

sage: X = Spec(QQ)
sage: X.identity_morphism()
Scheme endomorphism of Spectrum of Rational Field
  Defn: Identity map
point(v, check=True)
point_homset(S=None)

Return the set of S-valued points of this scheme.

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
point_set(S=None)

Return the set of S-valued points of this scheme.

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
structure_morphism()

Return the structure morphism from the scheme self to its base scheme.

EXAMPLES:

sage: A = AffineSpace(4, QQ)
sage: A.base_morphism()
Scheme morphism:
  From: Affine Space of dimension 4 over Rational Field
  To:   Spectrum of Rational Field
  Defn: Structure map
sage: X = Spec(QQ)
sage: X.base_morphism()
Scheme morphism:
  From: Spectrum of Rational Field
  To:   Spectrum of Integer Ring
  Defn: Structure map
union(X)

Return the disjoint union of the schemes self and X.

EXAMPLES:

sage: S = Spec(QQ)
sage: X = AffineSpace(1, QQ)
sage: S.union(X)
...
NotImplementedError
zeta_series(n, t)

Compute a power series approximation to the zeta function of a scheme over a finite field.

INPUT:

  • n - the number of terms of the power series to compute
  • t - the variable which the series should be returned

OUTPUT: A power series approximating the zeta function of self

EXAMPLES:

sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4,t)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)

Note that this function depends on count_points, which is only defined for prime order fields:

sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
...
NotImplementedError: Point counting only implemented for schemes over prime fields
sage.schemes.generic.scheme.is_AffineScheme(x)

Return True if x is an affine scheme.

EXAMPLES:

sage: from sage.schemes.generic.scheme import is_AffineScheme
sage: is_AffineScheme(5)
False
sage: E = Spec(QQ)
sage: is_AffineScheme(E)
True
sage.schemes.generic.scheme.is_Scheme(x)

Return True if x is a scheme.

EXAMPLES:

sage: from sage.schemes.generic.scheme import is_Scheme
sage: is_Scheme(5)
False
sage: X = Spec(QQ)
sage: is_Scheme(X)
True

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