36.11 Scheme morphism

Module: sage.schemes.generic.morphism

Scheme morphism

Author Log:

Module-level Functions

is_SchemeMorphism( f)

Class: PyMorphism

class PyMorphism
PyMorphism( self, parent)

Functions: category,$ \,$ codomain,$ \,$ domain,$ \,$ is_endomorphism

Special Functions: __init__,$ \,$ __pow__,$ \,$ _composition_,$ \,$ _repr_,$ \,$ _repr_defn,$ \,$ _repr_type

Class: SchemeMorphism

class SchemeMorphism
A scheme morphism
SchemeMorphism( self, parent)

Functions: glue_along_domains

glue_along_domains( self, other)

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

We construct a scheme isomorphic to the projective line over $ \Spec (\mathbf{Q})$ by gluing two copies of $ \mathbf{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 $ \mathbf{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

Special Functions: __init__,$ \,$ _repr_type

Class: SchemeMorphism_abelian_variety_coordinates_field

class SchemeMorphism_abelian_variety_coordinates_field

Special Functions: __mul__

Class: SchemeMorphism_affine_coordinates

class SchemeMorphism_affine_coordinates
A morphism determined by giving coordinates in a ring.

Input:

X
- a subscheme of an ambient affine space over a ring R.
v
- a list or tuple of coordinates in R

sage: A = AffineSpace(2, QQ)
sage: A(1,2)
(1, 2)
SchemeMorphism_affine_coordinates( self, X, v, [check=True])

Special Functions: __init__

Class: SchemeMorphism_coordinates

class SchemeMorphism_coordinates

Functions: scheme

Special Functions: __cmp__,$ \,$ __getitem__,$ \,$ __iter__,$ \,$ __tuple__,$ \,$ _latex_,$ \,$ _repr_

Class: SchemeMorphism_id

class SchemeMorphism_id
Return the identity morphism from X to itself.

sage: X = Spec(ZZ)
sage: X.identity_morphism()
Scheme endomorphism of Spectrum of Integer Ring
  Defn: Identity map
SchemeMorphism_id( self, X)

Special Functions: __init__,$ \,$ _repr_defn

Class: SchemeMorphism_on_points

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

Special Functions: __call__,$ \,$ _repr_defn

Class: SchemeMorphism_on_points_affine_space

class SchemeMorphism_on_points_affine_space
A morphism of schemes determined by rational functions that define what the morphism does on points in the ambient affine space.
SchemeMorphism_on_points_affine_space( self, parent, polys, [check=True])

Functions: defining_polynomials

Special Functions: __init__

Class: SchemeMorphism_on_points_projective_space

class SchemeMorphism_on_points_projective_space
A morphism of schemes determined by rational functions that define what the morphism does on points in the ambient projective space.
SchemeMorphism_on_points_projective_space( self, parent, polys, [check=True])

Functions: defining_polynomials

Special Functions: __init__

Class: SchemeMorphism_projective_coordinates_field

class SchemeMorphism_projective_coordinates_field
A morphism determined by giving coordinates in a field.

Input:

X
- a subscheme of an ambient projective space over a field K
v
- a list or tuple of coordinates in K

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

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
SchemeMorphism_projective_coordinates_field( self, X, v, [check=True])

Special Functions: __init__

Class: SchemeMorphism_projective_coordinates_ring

class SchemeMorphism_projective_coordinates_ring
A morphism determined by giving coordinates in a ring (how?).

SchemeMorphism_projective_coordinates_ring( self, X, v, [check=True])

Special Functions: __init__

Class: SchemeMorphism_spec

class SchemeMorphism_spec
A morphism of spectrums of rings

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
SchemeMorphism_spec( self, parent, phi, [check=True])

Functions: ring_homomorphism

Special Functions: __call__,$ \,$ __init__,$ \,$ _repr_defn,$ \,$ _repr_type

Class: SchemeMorphism_structure_map

class SchemeMorphism_structure_map
SchemeMorphism_structure_map( self, parent)

Input:

parent
- homset with codomain equal to the base scheme of the domain.

Special Functions: __init__,$ \,$ _repr_defn

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