# Schemes#

AUTHORS:

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

• Volker Braun (2011-08-11): documenting, improving, refactoring.

class sage.schemes.generic.scheme.AffineScheme(R, S=None, category=None)[source]#

Class for general affine schemes.

For affine spaces over a base ring and subschemes thereof, see sage.schemes.generic.algebraic_scheme.AffineSpace.

Element[source]#
base_extend(R)[source]#

Extend the base ring/scheme.

INPUT:

• R – an affine scheme or a commutative ring

EXAMPLES:

sage: Spec_ZZ = Spec(ZZ);  Spec_ZZ
Spectrum of Integer Ring
sage: Spec_ZZ.base_extend(QQ)
Spectrum of Rational Field

sage: Spec(ZZ['x']).base_extend(Spec(QQ))
Spectrum of Univariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> Spec_ZZ = Spec(ZZ);  Spec_ZZ
Spectrum of Integer Ring
>>> Spec_ZZ.base_extend(QQ)
Spectrum of Rational Field

>>> Spec(ZZ['x']).base_extend(Spec(QQ))
Spectrum of Univariate Polynomial Ring in x over Rational Field

coordinate_ring()[source]#

Return the underlying ring of this scheme.

OUTPUT:

A commutative ring.

EXAMPLES:

sage: Spec(QQ).coordinate_ring()
Rational Field
sage: Spec(PolynomialRing(QQ, 3, 'x')).coordinate_ring()
Multivariate Polynomial Ring in x0, x1, x2 over Rational Field

>>> from sage.all import *
>>> Spec(QQ).coordinate_ring()
Rational Field
>>> Spec(PolynomialRing(QQ, Integer(3), 'x')).coordinate_ring()
Multivariate Polynomial Ring in x0, x1, x2 over Rational Field

dimension()[source]#

Return the absolute dimension of this scheme.

OUTPUT:

Integer.

EXAMPLES:

sage: S = Spec(ZZ)
sage: S.dimension_absolute()
1
sage: S.dimension()
1

>>> from sage.all import *
>>> S = Spec(ZZ)
>>> S.dimension_absolute()
1
>>> S.dimension()
1

dimension_absolute()[source]#

Return the absolute dimension of this scheme.

OUTPUT:

Integer.

EXAMPLES:

sage: S = Spec(ZZ)
sage: S.dimension_absolute()
1
sage: S.dimension()
1

>>> from sage.all import *
>>> S = Spec(ZZ)
>>> S.dimension_absolute()
1
>>> S.dimension()
1

dimension_relative()[source]#

Return the relative dimension of this scheme over its base.

OUTPUT:

Integer.

EXAMPLES:

sage: S = Spec(ZZ)
sage: S.dimension_relative()
0

>>> from sage.all import *
>>> S = Spec(ZZ)
>>> S.dimension_relative()
0

hom(x, Y=None)[source]#

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

INPUT:

• x – anything that determines a scheme morphism; if x is a scheme, try to determine a natural map to x

• Y – the codomain scheme (optional); if Y is not given, try to determine Y from context

• check – boolean (default: True); whether to check the defining data for consistency

OUTPUT:

The scheme morphism from self to Y defined by x.

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: Natural morphism:
From: Integer Ring
To:   Rational Field

>>> from sage.all import *
>>> X = Spec(QQ)
>>> X.hom(ZZ.hom(QQ))
Affine Scheme morphism:
From: Spectrum of Rational Field
To:   Spectrum of Integer Ring
Defn: Natural morphism:
From: Integer Ring
To:   Rational Field

is_noetherian()[source]#

Return True if self is Noetherian, False otherwise.

EXAMPLES:

sage: Spec(ZZ).is_noetherian()
True

>>> from sage.all import *
>>> Spec(ZZ).is_noetherian()
True

class sage.schemes.generic.scheme.Scheme(X=None, category=None)[source]#

Bases: Parent

The base class for all schemes.

INPUT:

• X – a scheme, scheme morphism, commutative ring, commutative ring morphism, or None (optional). Determines the base scheme. If a commutative ring is passed, the spectrum of the ring will be used as base.

• category – the category (optional). Will be automatically constructed by default.

EXAMPLES:

sage: from sage.schemes.generic.scheme import Scheme
sage: Scheme(ZZ)
<sage.schemes.generic.scheme.Scheme_with_category object at ...>

>>> from sage.all import *
>>> from sage.schemes.generic.scheme import Scheme
>>> Scheme(ZZ)
<sage.schemes.generic.scheme.Scheme_with_category object at ...>


A scheme is in the category of all schemes over its base:

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

>>> from sage.all import *
>>> ProjectiveSpace(Integer(4), QQ).category()
Category of schemes over Rational Field


There is a special and unique $$\mathrm{Spec}(\ZZ)$$ that is the default base scheme:

sage: Spec(ZZ).base_scheme() is Spec(QQ).base_scheme()
True

>>> from sage.all import *
>>> Spec(ZZ).base_scheme() is Spec(QQ).base_scheme()
True

base_extend(Y)[source]#

Extend the base of the scheme.

Derived classes must override this method.

EXAMPLES:

sage: from sage.schemes.generic.scheme import Scheme
sage: X = Scheme(ZZ)
sage: X.base_scheme()
Spectrum of Integer Ring
sage: X.base_extend(QQ)
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> from sage.schemes.generic.scheme import Scheme
>>> X = Scheme(ZZ)
>>> X.base_scheme()
Spectrum of Integer Ring
>>> X.base_extend(QQ)
Traceback (most recent call last):
...
NotImplementedError

base_morphism()[source]#

Return the structure morphism from self to its base scheme.

OUTPUT:

A scheme morphism.

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

>>> from sage.all import *
>>> A = AffineSpace(Integer(4), QQ)
>>> A.base_morphism()
Scheme morphism:
From: Affine Space of dimension 4 over Rational Field
To:   Spectrum of Rational Field
Defn: Structure map

>>> X = Spec(QQ)
>>> X.base_morphism()
Scheme morphism:
From: Spectrum of Rational Field
To:   Spectrum of Integer Ring
Defn: Structure map

base_ring()[source]#

Return the base ring of the scheme self.

OUTPUT:

A commutative ring.

EXAMPLES:

sage: A = AffineSpace(4, QQ)
sage: A.base_ring()
Rational Field

sage: X = Spec(QQ)
sage: X.base_ring()
Integer Ring

>>> from sage.all import *
>>> A = AffineSpace(Integer(4), QQ)
>>> A.base_ring()
Rational Field

>>> X = Spec(QQ)
>>> X.base_ring()
Integer Ring

base_scheme()[source]#

Return the base scheme.

OUTPUT:

A scheme.

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

>>> from sage.all import *
>>> A = AffineSpace(Integer(4), QQ)
>>> A.base_scheme()
Spectrum of Rational Field

>>> X = Spec(QQ)
>>> X.base_scheme()
Spectrum of Integer Ring

coordinate_ring()[source]#

Return the coordinate ring.

OUTPUT:

The global coordinate ring of this scheme, if defined. Otherwise this raises 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)

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = (x**Integer(2) - y**Integer(2))*R
>>> X = Spec(R.quotient(I))
>>> X.coordinate_ring()
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
by the ideal (x^2 - y^2)

count_points(n)[source]#

Count points over finite fields.

INPUT:

• n – integer.

OUTPUT:

An integer. The number of points over $$\GF{q}, \ldots, \GF{q^n}$$ on a scheme over a finite field $$\GF{q}$$.

EXAMPLES:

sage: # needs sage.schemes
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)                              # needs sage.rings.finite_rings
[12, 96]

>>> from sage.all import *
>>> # needs sage.schemes
>>> P = PolynomialRing(GF(Integer(3)), names=('x',)); (x,) = P._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(3) + x**Integer(2) + Integer(1))
>>> C.count_points(Integer(4))
[6, 12, 18, 96]
>>> C.base_extend(GF(Integer(9),'a')).count_points(Integer(2))                              # needs sage.rings.finite_rings
[12, 96]

sage: P.<x,y,z> = ProjectiveSpace(GF(4, 't'), 2)                            # needs sage.rings.finite_rings
sage: X = P.subscheme([y^2*z - x^3 - z^3])                                  # needs sage.rings.finite_rings
sage: X.count_points(2)                                                     # needs sage.libs.singular sage.rings.finite_rings
[5, 17]

>>> from sage.all import *
>>> P = ProjectiveSpace(GF(Integer(4), 't'), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.finite_rings
>>> X = P.subscheme([y**Integer(2)*z - x**Integer(3) - z**Integer(3)])                                  # needs sage.rings.finite_rings
>>> X.count_points(Integer(2))                                                     # needs sage.libs.singular sage.rings.finite_rings
[5, 17]

dimension()[source]#

Return the absolute dimension of this scheme.

OUTPUT:

Integer.

EXAMPLES:

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

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = (x**Integer(2) - y**Integer(2))*R
>>> X = Spec(R.quotient(I))
>>> X.dimension_absolute()
Traceback (most recent call last):
...
NotImplementedError
>>> X.dimension()
Traceback (most recent call last):
...
NotImplementedError

dimension_absolute()[source]#

Return the absolute dimension of this scheme.

OUTPUT:

Integer.

EXAMPLES:

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

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = (x**Integer(2) - y**Integer(2))*R
>>> X = Spec(R.quotient(I))
>>> X.dimension_absolute()
Traceback (most recent call last):
...
NotImplementedError
>>> X.dimension()
Traceback (most recent call last):
...
NotImplementedError

dimension_relative()[source]#

Return the relative dimension of this scheme over its base.

OUTPUT:

Integer.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = (x^2 - y^2)*R
sage: X = Spec(R.quotient(I))
sage: X.dimension_relative()
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = (x**Integer(2) - y**Integer(2))*R
>>> X = Spec(R.quotient(I))
>>> X.dimension_relative()
Traceback (most recent call last):
...
NotImplementedError

hom(x, Y=None, check=True)[source]#

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

INPUT:

• x – anything that determines a scheme morphism; if x is a scheme, try to determine a natural map to x

• Y – the codomain scheme (optional); if Y is not given, try to determine Y from context

• check – boolean (default: True); whether to check the defining data for consistency

OUTPUT:

The scheme morphism from self to Y defined by x.

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

>>> from sage.all import *
>>> P = ProjectiveSpace(ZZ, Integer(3))
>>> 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()[source]#

Return the identity morphism.

OUTPUT:

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

>>> from sage.all import *
>>> X = Spec(QQ)
>>> X.identity_morphism()
Scheme endomorphism of Spectrum of Rational Field
Defn: Identity map

point(v, check=True)[source]#

Create a point.

INPUT:

• v – anything that defines a point

• check – boolean (default: True); whether to check the defining data for consistency

OUTPUT:

A point of the scheme.

EXAMPLES:

sage: A2 = AffineSpace(QQ, 2)
sage: A2.point([4, 5])
(4, 5)

sage: R.<t> = PolynomialRing(QQ)
sage: E = EllipticCurve([t + 1, t, t, 0, 0])                                # needs sage.schemes
sage: E.point([0, 0])                                                       # needs sage.schemes
(0 : 0 : 1)

>>> from sage.all import *
>>> A2 = AffineSpace(QQ, Integer(2))
>>> A2.point([Integer(4), Integer(5)])
(4, 5)

>>> R = PolynomialRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> E = EllipticCurve([t + Integer(1), t, t, Integer(0), Integer(0)])                                # needs sage.schemes
>>> E.point([Integer(0), Integer(0)])                                                       # needs sage.schemes
(0 : 0 : 1)

point_homset(S=None)[source]#

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

INPUT:

• S – a commutative ring.

OUTPUT:

The set of morphisms $$\mathrm{Spec}(S) \to X$$.

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

>>> from sage.all import *
>>> P = ProjectiveSpace(ZZ, Integer(3))
>>> P.point_homset(ZZ)
Set of rational points of Projective Space of dimension 3 over Integer Ring
>>> P.point_homset(QQ)
Set of rational points of Projective Space of dimension 3 over Rational Field
>>> P.point_homset(GF(Integer(11)))
Set of rational points of Projective Space of dimension 3 over
Finite Field of size 11

point_set(S=None)[source]#

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

INPUT:

• S – a commutative ring.

OUTPUT:

The set of morphisms $$\mathrm{Spec}(S) \to X$$.

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

>>> from sage.all import *
>>> P = ProjectiveSpace(ZZ, Integer(3))
>>> P.point_homset(ZZ)
Set of rational points of Projective Space of dimension 3 over Integer Ring
>>> P.point_homset(QQ)
Set of rational points of Projective Space of dimension 3 over Rational Field
>>> P.point_homset(GF(Integer(11)))
Set of rational points of Projective Space of dimension 3 over
Finite Field of size 11

structure_morphism()[source]#

Return the structure morphism from self to its base scheme.

OUTPUT:

A scheme morphism.

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

>>> from sage.all import *
>>> A = AffineSpace(Integer(4), QQ)
>>> A.base_morphism()
Scheme morphism:
From: Affine Space of dimension 4 over Rational Field
To:   Spectrum of Rational Field
Defn: Structure map

>>> X = Spec(QQ)
>>> X.base_morphism()
Scheme morphism:
From: Spectrum of Rational Field
To:   Spectrum of Integer Ring
Defn: Structure map

union(X)[source]#

Return the disjoint union of the schemes self and X.

EXAMPLES:

sage: S = Spec(QQ)
sage: X = AffineSpace(1, QQ)
sage: S.union(X)
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> S = Spec(QQ)
>>> X = AffineSpace(Integer(1), QQ)
>>> S.union(X)
Traceback (most recent call last):
...
NotImplementedError

zeta_function()[source]#

Compute the zeta function of a generic scheme.

Derived classes should override this method.

OUTPUT: rational function in one variable.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(GF(4, 't'), 2)                            # needs sage.rings.finite_rings
sage: X = P.subscheme([y^2*z - x^3 - z^3])                                  # needs sage.rings.finite_rings
sage: X.zeta_function()                                                     # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> P = ProjectiveSpace(GF(Integer(4), 't'), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.finite_rings
>>> X = P.subscheme([y**Integer(2)*z - x**Integer(3) - z**Integer(3)])                                  # needs sage.rings.finite_rings
>>> X.zeta_function()                                                     # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError

zeta_series(n, t)[source]#

Return the zeta series.

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)                                 # needs sage.schemes
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4, t)                                                   # needs sage.schemes
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)

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(3)), names=('x',)); (x,) = P._first_ngens(1)
>>> C = HyperellipticCurve(x**Integer(3) + x**Integer(2) + Integer(1))                                 # needs sage.schemes
>>> R = PowerSeriesRing(Integers(), names=('t',)); (t,) = R._first_ngens(1)
>>> C.zeta_series(Integer(4), t)                                                   # needs sage.schemes
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
>>> (Integer(1)+Integer(2)*t+Integer(3)*t**Integer(2))/(Integer(1)-t)/(Integer(1)-Integer(3)*t) + O(t**Integer(5))
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)


If the scheme has a method zeta_function, this is used to provide the required approximation. Otherwise this function depends on count_points, which is only defined for prime order fields for general schemes. Nonetheless, since Issue #15108 and Issue #15148, it supports hyperelliptic curves over non-prime fields:

sage: C.base_extend(GF(9, 'a')).zeta_series(4, t)                           # needs sage.rings.finite_rings sage.schemes
1 + 12*t + 120*t^2 + 1092*t^3 + 9840*t^4 + O(t^5)

>>> from sage.all import *
>>> C.base_extend(GF(Integer(9), 'a')).zeta_series(Integer(4), t)                           # needs sage.rings.finite_rings sage.schemes
1 + 12*t + 120*t^2 + 1092*t^3 + 9840*t^4 + O(t^5)

sage: P.<x,y,z> = ProjectiveSpace(GF(4, 't'), 2)                            # needs sage.rings.finite_rings
sage: X = P.subscheme([y^2*z - x^3 - z^3])                                  # needs sage.rings.finite_rings

sage: R.<t> = PowerSeriesRing(Integers())
sage: X.zeta_series(2, t)                                                   # needs sage.libs.singular sage.rings.finite_rings
1 + 5*t + 21*t^2 + O(t^3)

>>> from sage.all import *
>>> P = ProjectiveSpace(GF(Integer(4), 't'), Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)# needs sage.rings.finite_rings
>>> X = P.subscheme([y**Integer(2)*z - x**Integer(3) - z**Integer(3)])                                  # needs sage.rings.finite_rings

>>> R = PowerSeriesRing(Integers(), names=('t',)); (t,) = R._first_ngens(1)
>>> X.zeta_series(Integer(2), t)                                                   # needs sage.libs.singular sage.rings.finite_rings
1 + 5*t + 21*t^2 + O(t^3)

sage.schemes.generic.scheme.is_AffineScheme(x)[source]#

Return True if $$x$$ is an affine scheme.

EXAMPLES:

sage: from sage.schemes.generic.scheme import is_AffineScheme
sage: is_AffineScheme(5)
doctest:warning...
DeprecationWarning: The function is_AffineScheme is deprecated; use 'isinstance(..., AffineScheme)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
False
sage: E = Spec(QQ)
sage: is_AffineScheme(E)
True

>>> from sage.all import *
>>> from sage.schemes.generic.scheme import is_AffineScheme
>>> is_AffineScheme(Integer(5))
doctest:warning...
DeprecationWarning: The function is_AffineScheme is deprecated; use 'isinstance(..., AffineScheme)' instead.
See https://github.com/sagemath/sage/issues/38022 for details.
False
>>> E = Spec(QQ)
>>> is_AffineScheme(E)
True

sage.schemes.generic.scheme.is_Scheme(x)[source]#

Test whether x is a scheme.

INPUT:

• x – anything.

OUTPUT:

Boolean. Whether x derives from Scheme.

EXAMPLES:

sage: from sage.schemes.generic.scheme import is_Scheme
sage: is_Scheme(5)
doctest:warning...
DeprecationWarning: The function is_Scheme is deprecated; use 'isinstance(..., Scheme)' or categories instead.
See https://github.com/sagemath/sage/issues/38022 for details.
False
sage: X = Spec(QQ)
sage: is_Scheme(X)
True

>>> from sage.all import *
>>> from sage.schemes.generic.scheme import is_Scheme
>>> is_Scheme(Integer(5))
doctest:warning...
DeprecationWarning: The function is_Scheme is deprecated; use 'isinstance(..., Scheme)' or categories instead.
See https://github.com/sagemath/sage/issues/38022 for details.
False
>>> X = Spec(QQ)
>>> is_Scheme(X)
True