23.2 Schemes

Module: sage.schemes.generic.scheme

Author Log:

Module-level Functions

is_AffineScheme( x)

Return True if $ x$ is an affine scheme.

sage: is_AffineScheme(5)
False
sage: E = Spec(QQ)
sage: is_AffineScheme(E)
True

is_Scheme( x)

Return True if $ x$ is a scheme.

sage: is_Scheme(5)
False
sage: X = Spec(QQ)
sage: is_Scheme(X)
True

Class: AffineScheme

class AffineScheme
An abstract affine scheme.

Functions: hom

hom( self, x, [Y=None])

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

If Y is not given, try to determine from context.

We construct the inclusion from $ \Spec (\mathbf{Q})$ into $ \Spec (\mathbf{Z})$ induced by the inclusion from $ \mathbf{Z}$ into $ \mathbf{Q}$ .

sage: X = Spec(Q)
sage: X.hom(Z.hom(Q))
Affine Scheme morphism:
  From: Spectrum of Rational Field
  To:   Spectrum of Integer Ring
  Defn: Coercion morphism:
          From: Integer Ring
          To:   Rational Field

Class: Scheme

class Scheme
A scheme.
Scheme( self, X)

Functions: base_extend,$  $ base_morphism,$  $ base_ring,$  $ base_scheme,$  $ category,$  $ coordinate_ring,$  $ dimension,$  $ hom,$  $ identity_morphism,$  $ point,$  $ point_homset,$  $ point_set,$  $ structure_morphism

base_extend( self, Y)

Y is either a scheme in the same category as self or a ring.

coordinate_ring( self)

Return the coordinate ring of this scheme, if defined. Otherwise raise a ValueError.

dimension( self)

Return the relative dimension of this scheme over its base.

hom( self, 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.

point_set( self, S)

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

structure_morphism( self)

Same as self.base_morphism().

Special Functions: __add__,$  $ __call__,$  $ __cmp__,$  $ __div__,$  $ _Hom_,$  $ _homset_class,$  $ _point_class,$  $ _point_morphism_class

__call__( self)

If S is a ring or scheme, return the set $ X(S)$ of $ S$ -valued points on $ X$ . If $ S$ is a list or tuple or just the coordinates, return a point in $ X(T)$ , where $ T$ is the base scheme of self.

sage: A = AffineSpace(2, QQ)

We create some point sets:

sage: A(QQ)
Set of Rational Points of Affine Space of dimension 2 over Rational Field  

sage: A(RR)
Set of Rational Points over Real Field with 53 bits of precision of Affine
Space of dimension 2 over Rational Field
sage: A(NumberField(x^2+1))
Set of Rational Points over Number Field in a with defining polynomial x^2
+ 1
of Affine Space of dimension 2 over Rational Field
sage: A(GF(7))
Traceback (most recent call last):
...
ValueError: There must be a natural map S --> R, but S = Rational Field and
R = Finite Field of size 7

We create some points:

sage: A(QQ)([1,0])
(1, 0)

We create the same point by giving the coordinates of the point directly.

sage: A( 1,0 )
(1, 0)

__div__( self, Y)

Return the base extension of self to Y.

sage: A = AffineSpace(3, Z)
sage: A
Affine Space of dimension 3 over Integer Ring
sage: A/Q
Affine Space of dimension 3 over Rational Field
sage: A/GF(7)
Affine Space of dimension 3 over Finite Field of size 7

_Hom_( self, Y, [cat=True], [check=None])

Return the set of scheme morphisms from self to Y.

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