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.
  • Ben Hutz (June 2012): added support for projective ring
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: X = Spec(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'>
change_ring(R, check=True)

Returns a new SchemeMorphism_point which is self coerced to R. If \(check\) is true, then the initialization checks are performed.

INPUT:

  • R – a ring
  • check – Boolean (optional)

OUTPUT:

EXAMPLES:

sage: P.<x,y,z>=ProjectiveSpace(ZZ,2)
sage: X=P.subscheme(x^2-y^2)
sage: X(23,23,1).change_ring(GF(13))
(10 : 10 : 1)
sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: P(-2/3,1).change_ring(CC)
(-0.666666666666667 : 1.00000000000000)
sage: P.<x,y>=ProjectiveSpace(ZZ,1)
sage: P(152,113).change_ring(Zp(5))
(2 + 5^2 + 5^3 + O(5^20) : 3 + 2*5 + 4*5^2 + O(5^20))
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_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: 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
base_ring()

Return the base ring of self, that is, the ring over which the coefficients of self is given as polynomials.

OUTPUT:

  • ring

EXAMPLES:

sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: H=Hom(P,P)
sage: f=H([3/5*x^2,6*y^2])
sage: f.base_ring()
Rational Field
sage: R.<t>=PolynomialRing(ZZ,1)
sage: P.<x,y>=ProjectiveSpace(R,1)
sage: H=Hom(P,P)
sage: f=H([3*x^2,y^2])
sage: f.base_ring()
Multivariate Polynomial Ring in t over Integer Ring
change_ring(R, check=True)

Returns a new SchemeMorphism_polynomial which is self coerced to \(R\). If check is True, then the initialization checks are performed.

INPUT:

  • R – ring
  • check – Boolean

OUTPUT:

  • element of the coordinate ring of the domain

EXAMPLES:

sage: P.<x,y>=ProjectiveSpace(ZZ,1)
sage: H=Hom(P,P)
sage: f=H([3*x^2,y^2])
sage: f.change_ring(GF(3))
Traceback (most recent call last):
...
ValueError: polys (=[0, y^2]) must be of the same degree
sage: P.<x,y,z>=ProjectiveSpace(QQ,2)
sage: H=Hom(P,P)
sage: f=H([5/2*x^3 + 3*x*y^2-y^3,3*z^3 + y*x^2, x^3-z^3])
sage: f.change_ring(GF(3))
Scheme endomorphism of Projective Space of dimension 2 over Finite Field of size 3
  Defn: Defined on coordinates by sending (x : y : z) to
        (x^3 - y^3 : x^2*y : x^3 - z^3)
sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: X=P.subscheme([5*x^2-y^2])
sage: H=Hom(X,X)
sage: f=H([x,y])
sage: f.change_ring(GF(3))
Scheme endomorphism of Closed subscheme of Projective Space of dimension
1 over Finite Field of size 3 defined by:
  -x^2 - y^2
  Defn: Defined on coordinates by sending (x : y) to
        (x : y)
coordinate_ring()

Returns the coordinate ring of the ambient projective space the multivariable polynomial ring over the base ring

OUTPUT:

  • ring

EXAMPLES:

sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: H=Hom(P,P)
sage: f=H([3/5*x^2,6*y^2])
sage: f.coordinate_ring()
Multivariate Polynomial Ring in x, y over Rational Field
sage: R.<t>=PolynomialRing(ZZ,1)
sage: P.<x,y>=ProjectiveSpace(R,1)
sage: H=Hom(P,P)
sage: f=H([3*x^2,y^2])
sage: f.coordinate_ring()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring
in t over Integer Ring
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_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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