Scheme morphism¶

Note

You should never create the morphisms directy. Instead, use the hom() and Hom() methods that are inherited by all schemes.

If you want to extend the Sage library with some new kind of scheme, your new class (say, myscheme) should provide a method

myscheme._morphism(*args, **kwds) returning a morphism between two schemes in your category, usually defined via polynomials. Your morphism class should derive from SchemeMorphism_polynomial. These morphisms will usually be elements of the Hom-set SchemeHomset_generic.

Optionally, you can also provide a special Hom-set class for your subcategory of schemes. If you want to do this, you should also provide a method

myscheme._homset(*args, **kwds) returning a Hom-set, which must be an element of a derived class of $class:`~sage.schemes.generic.homset.SchemeHomset_generic$ . If your new Hom-set class does not use myscheme._morphism then you do not have to provide it.

Note that points on schemes are morphisms $Spec(K)\to X$ , too. But we typically use a different notation, so they are implemented in a different derived class. For this, you should implement a method

myscheme._point(*args, **kwds) returning a point, that is, a morphism $Spec(K)\to X$ . Your point class should derive from SchemeMorphism_point.

Optionally, you can also provide a special Hom-set for the points, for example the point Hom-set can provide a method to enumerate all points. If you want to do this, you should also provide a method

myscheme._point_homset(*args, **kwds) returning the homset of points. The Hom-sets of points are implemented in classes named SchemeHomset_points_.... If your new Hom-set class does not use myscheme._point then you do not have to provide it.

AUTHORS:

David Kohel, William Stein
William Stein (2006-02-11): fixed bug where P(0,0,0) was allowed as a projective point.
Volker Braun (2011-08-08): Renamed classes, more documentation, misc cleanups.

class sage.schemes.generic.morphism.SchemeMorphism(parent)¶

Bases: sage.structure.element.Element

Base class for scheme morphisms

INPUT:

parent – the parent of the morphism.

EXAMPLES:

sage: from sage.schemes.generic.scheme import Scheme
sage: X = Scheme(ZZ)
sage: Hom = X.Hom(X)
sage: from sage.schemes.generic.morphism import SchemeMorphism
sage: f = SchemeMorphism(Hom)
sage: type(f)
<class 'sage.schemes.generic.morphism.SchemeMorphism'>

category()¶

Return the category of the Hom-set.

OUTPUT:

A category.

EXAMPLES:

sage: A2 = AffineSpace(QQ,2)
sage: A2.structure_morphism().category()
Category of hom sets in Category of Schemes

codomain()¶

Return the codomain (range) of the morphism.

OUTPUT:

A scheme. The codomain of the morphism self.

EXAMPLES:

sage: A2 = AffineSpace(QQ,2)
sage: A2.structure_morphism().codomain()
Spectrum of Rational Field

domain()¶

Return the domain of the morphism.

OUTPUT:

A scheme. The domain of the morphism self.

EXAMPLES:

sage: A2 = AffineSpace(QQ,2)
sage: A2.structure_morphism().domain()
Affine Space of dimension 2 over Rational Field

glue_along_domains(other)¶

Glue two morphism

INPUT:

other – a scheme morphism with the same domain.

OUTPUT:

Assuming that self and other are open immersions with the same domain, return scheme obtained by gluing along the images.

EXAMPLES:

We construct a scheme isomorphic to the projective line over $\mathrm{Spec}(\QQ)$ by gluing two copies of $\mathbb{A}^1$ minus a point:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S.<xbar, ybar> = R.quotient(x*y - 1)
sage: Rx = PolynomialRing(QQ, 'x')
sage: i1 = Rx.hom([xbar])
sage: Ry = PolynomialRing(QQ, 'y')
sage: i2 = Ry.hom([ybar])
sage: Sch = Schemes()
sage: f1 = Sch(i1)
sage: f2 = Sch(i2)

Now f1 and f2 have the same domain, which is a $\mathbb{A}^1$ minus a point. We glue along the domain:

sage: P1 = f1.glue_along_domains(f2)
sage: P1
Scheme obtained by gluing X and Y along U, where
  X: Spectrum of Univariate Polynomial Ring in x over Rational Field
  Y: Spectrum of Univariate Polynomial Ring in y over Rational Field
  U: Spectrum of Quotient of Multivariate Polynomial Ring in x, y 
  over Rational Field by the ideal (x*y - 1)

sage: a, b = P1.gluing_maps()
sage: a
Affine Scheme morphism:
 From: Spectrum of Quotient of Multivariate Polynomial Ring in x, y 
       over Rational Field by the ideal (x*y - 1)
  To:   Spectrum of Univariate Polynomial Ring in x over Rational Field
  Defn: Ring morphism:
          From: Univariate Polynomial Ring in x over Rational Field
          To:   Quotient of Multivariate Polynomial Ring in x, y over 
                Rational Field by the ideal (x*y - 1)
          Defn: x |--> xbar
sage: b
Affine Scheme morphism:
  From: Spectrum of Quotient of Multivariate Polynomial Ring in x, y 
        over Rational Field by the ideal (x*y - 1)
  To:   Spectrum of Univariate Polynomial Ring in y over Rational Field
  Defn: Ring morphism:
          From: Univariate Polynomial Ring in y over Rational Field
          To:   Quotient of Multivariate Polynomial Ring in x, y over 
                Rational Field by the ideal (x*y - 1)
          Defn: y |--> ybar

is_endomorphism()¶

Return wether the morphism is an endomorphism.

OUTPUT:

Boolean. Whether the domain and codomain are identical.

EXAMPLES:

sage: X = AffineSpace(QQ,2)
sage: X.structure_morphism().is_endomorphism()
False
sage: X.identity_morphism().is_endomorphism()
True

class sage.schemes.generic.morphism.SchemeMorphism_id(X)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism

Return the identity morphism from $X$ to itself.

INPUT:

X – the scheme.

EXAMPLES:

sage: X = Spec(ZZ)
sage: X.identity_morphism()  # indirect doctest
Scheme endomorphism of Spectrum of Integer Ring
  Defn: Identity map

class sage.schemes.generic.morphism.SchemeMorphism_point(parent)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism

Base class for rational points on schemes.

Recall that the $K$ -rational points of a scheme $X$ over $k$ can be identified with the set of morphisms $Spec(K) o X$ . In Sage, the rational points are implemented by such scheme morphisms.

EXAMPLES:

sage: from sage.schemes.generic.morphism import SchemeMorphism
sage: f = SchemeMorphism(Spec(ZZ).Hom(Spec(ZZ)))
sage: type(f)
<class 'sage.schemes.generic.morphism.SchemeMorphism'>

scheme()¶

Return the scheme whose point is represented.

OUTPUT:

A scheme.

EXAMPLES:

sage: A = AffineSpace(2, QQ)
sage: a = A(1,2)
sage: a.scheme()
Affine Space of dimension 2 over Rational Field

class sage.schemes.generic.morphism.SchemeMorphism_point_abelian_variety_field(X, v, check=True)¶

Bases: sage.structure.element.AdditiveGroupElement, sage.schemes.generic.morphism.SchemeMorphism_point_projective_field

A rational point of an abelian variety over a field.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: origin = E(0)
sage: origin.domain()
Spectrum of Rational Field
sage: origin.codomain()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

class sage.schemes.generic.morphism.SchemeMorphism_point_affine(X, v, check=True)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism_point

A rational point on an affine scheme.

INPUT:

X – a subscheme of an ambient affine space over a ring $R$ .
v – a list/tuple/iterable of coordinates in $R$ .
check – boolean (optional, default:True). Whether to check the input for consistency.

EXAMPLES:

sage: A = AffineSpace(2, QQ)
sage: A(1,2)
(1, 2)

class sage.schemes.generic.morphism.SchemeMorphism_point_projective_field(X, v, check=True)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism_point_projective_ring

A rational point of projective space over a field.

INPUT:

X – a subscheme of an ambient projective space over a field $K$
v – a list or tuple of coordinates in $K$
check – boolean (optional, default:True). Whether to check the input for consistency.

EXAMPLES:

sage: P = ProjectiveSpace(3, RR)
sage: P(2,3,4,5)
(0.400000000000000 : 0.600000000000000 : 0.800000000000000 : 1.00000000000000)

Not all homogeneous coordinates are allowed to vanish simultaneously:

sage: P = ProjectiveSpace(3, QQ)
sage: P(0,0,0,0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0

class sage.schemes.generic.morphism.SchemeMorphism_point_projective_ring(X, v, check=True)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism_point

A rational point of projective space over a ring (how?).

Currently this is not implemented.

EXAMPLES:

sage: from sage.schemes.generic.morphism import SchemeMorphism_point_projective_ring sage: SchemeMorphism_point_projective_ring(None, None) Traceback (most recent call last): ... NotImplementedError

class sage.schemes.generic.morphism.SchemeMorphism_polynomial(parent, polys, check=True)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism

A morphism of schemes determined by polynomials that define what the morphism does on points in the ambient space.

INPUT:

parent – Hom-set whose domain and codomain are affine schemes.
polys – a list/tuple/iterable of polynomials defining the scheme morphism.
check – boolean (optional, default:True). Whether to check the input for consistency.

EXAMPLES:

An example involving the affine plane:

sage: R.<x,y> = QQ[]
sage: A2 = AffineSpace(R)
sage: H = A2.Hom(A2)
sage: f = H([x-y, x*y])
sage: f([0,1])
(-1, 0)

An example involving the projective line:

sage: R.<x,y> = QQ[]
sage: P1 = ProjectiveSpace(R)
sage: H = P1.Hom(P1)
sage: f = H([x^2+y^2,x*y])
sage: f([0,1])
(1 : 0)

Some checks are performed to make sure the given polynomials define a morphism:

sage: R.<x,y> = QQ[]
sage: P1 = ProjectiveSpace(R)
sage: H = P1.Hom(P1)
sage: f = H([x^2, x*y])
Traceback (most recent call last):
...
ValueError: polys (=[x^2, x*y]) must not have common factors

sage: f = H([exp(x),exp(y)])
Traceback (most recent call last):
...
TypeError: polys (=[e^x, e^y]) must be elements of
Multivariate Polynomial Ring in x, y over Rational Field

defining_polynomials()¶

Return the defining polynomials.

OUTPUT:

An immutable sequence of polynomials that defines this scheme morphism.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: A.<x,y> = AffineSpace(R)
sage: H = A.Hom(A)
sage: H([x^3+y, 1-x-y]).defining_polynomials()
[x^3 + y, -x - y + 1]

class sage.schemes.generic.morphism.SchemeMorphism_polynomial_affine_space(parent, polys, check=True)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism_polynomial

A morphism of schemes determined by rational functions that define what the morphism does on points in the ambient affine space.

EXAMPLES:

sage: RA.<x,y> = QQ[]
sage: A2 = AffineSpace(RA)
sage: RP.<u,v,w> = QQ[]
sage: P2 = ProjectiveSpace(RP)
sage: H = A2.Hom(P2)
sage: f = H([x, y, 1])
sage: f
Scheme morphism:
  From: Affine Space of dimension 2 over Rational Field
  To:   Projective Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to
        (x : y : 1)

class sage.schemes.generic.morphism.SchemeMorphism_polynomial_projective_space(parent, polys, check=True)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism_polynomial

A morphism of schemes determined by rational functions that define what the morphism does on points in the ambient projective space.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: P1 = ProjectiveSpace(R)
sage: H = P1.Hom(P1)
sage: H([y,2*x])
Scheme endomorphism of Projective Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x : y) to
        (y : 2*x)

An example of a morphism between projective plane curves (see #10297):

sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: f = x^3+y^3+60*z^3
sage: g = y^2*z-( x^3 - 6400*z^3/3)
sage: C = Curve(f)
sage: E = Curve(g)
sage: xbar,ybar,zbar = C.coordinate_ring().gens()
sage: H = C.Hom(E)
sage: H([zbar,xbar-ybar,-(xbar+ybar)/80])
Scheme morphism:
  From: Projective Curve over Rational Field defined by x^3 + y^3 + 60*z^3
  To:   Projective Curve over Rational Field defined by -x^3 + y^2*z + 6400/3*z^3
  Defn: Defined on coordinates by sending (x : y : z) to
        (z : x - y : -1/80*x - 1/80*y)

A more complicated example:

sage: P2.<x,y,z> = ProjectiveSpace(2,QQ)
sage: P1 = P2.subscheme(x-y)
sage: H12 = P1.Hom(P2)
sage: H12([x^2,x*z, z^2])
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x - y
To:   Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
      (y^2 : y*z : z^2)

We illustrate some error checking:

sage: R.<x,y> = QQ[]
sage: P1 = ProjectiveSpace(R)
sage: H = P1.Hom(P1)
sage: f = H([x-y, x*y])
Traceback (most recent call last):
...
ValueError: polys (=[x - y, x*y]) must be of the same degree

sage: H([x-1, x*y+x])
Traceback (most recent call last):
...
ValueError: polys (=[x - 1, x*y + x]) must be homogeneous

sage: H([exp(x),exp(y)])
Traceback (most recent call last):
...
TypeError: polys (=[e^x, e^y]) must be elements of 
Multivariate Polynomial Ring in x, y over Rational Field

class sage.schemes.generic.morphism.SchemeMorphism_spec(parent, phi, check=True)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism

Morphism of spectra of rings

INPUT:

parent – Hom-set whose domain and codomain are affine schemes.
phi – a ring morphism with matching domain and codomain.
check – boolean (optional, default:True). Whether to check the input for consistency.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: phi = R.hom([QQ(7)]); phi
Ring morphism:
  From: Univariate Polynomial Ring in x over Rational Field
  To:   Rational Field
  Defn: x |--> 7

sage: X = Spec(QQ); Y = Spec(R)
sage: f = X.hom(phi); f
Affine Scheme morphism:
  From: Spectrum of Rational Field
  To:   Spectrum of Univariate Polynomial Ring in x over Rational Field
  Defn: Ring morphism:
          From: Univariate Polynomial Ring in x over Rational Field
          To:   Rational Field
          Defn: x |--> 7

sage: f.ring_homomorphism()
Ring morphism:
  From: Univariate Polynomial Ring in x over Rational Field
  To:   Rational Field
  Defn: x |--> 7

ring_homomorphism()¶

Return the underlying ring homomorphism.

OUTPUT:

A ring homomorphism.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: phi = R.hom([QQ(7)])
sage: X = Spec(QQ); Y = Spec(R)
sage: f = X.hom(phi)
sage: f.ring_homomorphism()
Ring morphism:
  From: Univariate Polynomial Ring in x over Rational Field
  To:   Rational Field
  Defn: x |--> 7

class sage.schemes.generic.morphism.SchemeMorphism_structure_map(parent)¶

Bases: sage.schemes.generic.morphism.SchemeMorphism

The structure morphism

INPUT:

parent – Hom-set with codomain equal to the base scheme of the domain.

EXAMPLES:

sage: Spec(ZZ).structure_morphism()    # indirect doctest
Scheme morphism:
  From: Spectrum of Integer Ring
  To:   Spectrum of Integer Ring
  Defn: Structure map

sage.schemes.generic.morphism.is_SchemeMorphism(f)¶

Test whether f is a scheme morphism.

INPUT:

f – anything.

OUTPUT:

Boolean. Return True if f is a scheme morphism or a point on an elliptic curve.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ,2); H = A.Hom(A)
sage: f = H([y,x^2+y]); f
Scheme endomorphism of Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to
        (y, x^2 + y)
sage: from sage.schemes.generic.morphism import is_SchemeMorphism
sage: is_SchemeMorphism(f)
True

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