Schemes¶

AUTHORS:

William Stein, David Kohel, Kiran Kedlaya (2008): added zeta_series

class sage.schemes.generic.scheme.AffineScheme(X, category=None)¶

Bases: sage.schemes.generic.scheme.Scheme

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, category=None)¶

Bases: sage.structure.parent.Parent

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

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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