Coverage for local/lib/python2.7/site-packages/sage/schemes/generic/point.py : 30%

""" 

Points on schemes 

""" 

#******************************************************************************* 

# Copyright (C) 2006 William Stein 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#******************************************************************************* 

from sage.structure.element import Element 

from sage.structure.richcmp import richcmp 

######################################################## 

# Base class for points on a scheme, either topological 

# or defined by a morphism. 

######################################################## 

class SchemePoint(Element): 

""" 

Base class for points on a scheme, either topological or defined 

by a morphism. 

""" 

def __init__(self, S, parent=None): 

""" 

INPUT: 

- ``S`` -- a scheme 

- ``parent`` -- the parent in which to construct this point 

TESTS:: 

sage: from sage.schemes.generic.point import SchemePoint 

sage: S = Spec(ZZ) 

sage: P = SchemePoint(S); P 

Point on Spectrum of Integer Ring 

""" 

Element.__init__(self, parent) 

self.__S = S 

def scheme(self): 

""" 

Return the scheme on which self is a point. 

EXAMPLES:: 

sage: from sage.schemes.generic.point import SchemePoint 

sage: S = Spec(ZZ) 

sage: P = SchemePoint(S) 

sage: P.scheme() 

Spectrum of Integer Ring 

""" 

return self.__S 

def _repr_(self): 

""" 

Return a string representation of this generic scheme point. 

TESTS:: 

sage: from sage.schemes.generic.point import SchemePoint 

sage: S = Spec(ZZ) 

sage: P = SchemePoint(S); P 

Point on Spectrum of Integer Ring 

sage: P._repr_() 

'Point on Spectrum of Integer Ring' 

""" 

return "Point on %s"%self.__S 

######################################################## 

# Topological points on a scheme 

######################################################## 

def is_SchemeTopologicalPoint(x): 

return isinstance(x, SchemeTopologicalPoint) 

class SchemeTopologicalPoint(SchemePoint): 

""" 

Base class for topological points on schemes. 

""" 

def __init__(self, S): 

""" 

INPUT: 

- ``S`` -- a scheme 

TESTS: 

The parent of a topological point is the scheme on which it 

lies (see :trac:`7946`):: 

sage: R = Zmod(8) 

sage: S = Spec(R) 

sage: x = S(R.ideal(2)) 

sage: isinstance(x, sage.schemes.generic.point.SchemeTopologicalPoint) 

True 

sage: x.parent() is S 

True 

""" 

SchemePoint.__init__(self, S, parent=S) 

class SchemeTopologicalPoint_affine_open(SchemeTopologicalPoint): 

def __init__(self, u, x): 

""" 

INPUT: 

- ``u`` -- morphism with domain an affine scheme `U` 

- ``x`` -- topological point on `U` 

""" 

SchemeTopologicalPoint.__init__(self, u.codomain()) 

self.__u = u 

self.__x = x 

def _repr_(self): 

return "Point on %s defined by x in U, where:\n U: %s\n x: %s"%(\ 

self.scheme(), self.embedding_of_affine_open().domain(), 

self.point_on_affine()) 

def point_on_affine(self): 

""" 

Return the scheme point on the affine open U. 

""" 

return self.__x 

def affine_open(self): 

""" 

Return the affine open subset U. 

""" 

return self.__u.domain() 

def embedding_of_affine_open(self): 

""" 

Return the embedding from the affine open subset U into this 

scheme. 

""" 

return self.__u 

class SchemeTopologicalPoint_prime_ideal(SchemeTopologicalPoint): 

def __init__(self, S, P, check=False): 

""" 

INPUT: 

- ``S`` -- an affine scheme 

- ``P`` -- a prime ideal of the coordinate ring of `S`, or 

anything that can be converted into such an ideal 

TESTS:: 

sage: from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal 

sage: S = Spec(ZZ) 

sage: P = SchemeTopologicalPoint_prime_ideal(S, 3); P 

Point on Spectrum of Integer Ring defined by the Principal ideal (3) of Integer Ring 

sage: SchemeTopologicalPoint_prime_ideal(S, 6, check=True) 

Traceback (most recent call last): 

... 

ValueError: The argument Principal ideal (6) of Integer Ring must be a prime ideal of Integer Ring 

sage: SchemeTopologicalPoint_prime_ideal(S, ZZ.ideal(7)) 

Point on Spectrum of Integer Ring defined by the Principal ideal (7) of Integer Ring 

We define a parabola in the projective plane as a point 

corresponding to a prime ideal:: 

sage: P2.<x, y, z> = ProjectiveSpace(2, QQ) 

sage: SchemeTopologicalPoint_prime_ideal(P2, y*z-x^2) 

Point on Projective Space of dimension 2 over Rational Field defined by the Ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field 

""" 

R = S.coordinate_ring() 

from sage.rings.ideal import is_Ideal 

if not is_Ideal(P): 

P = R.ideal(P) 

elif P.ring() is not R: 

P = R.ideal(P.gens()) 

# ideally we would have check=True by default, but 

# unfortunately is_prime() is only implemented in a small 

# number of cases 

if check and not P.is_prime(): 

raise ValueError("The argument %s must be a prime ideal of %s"%(P, R)) 

SchemeTopologicalPoint.__init__(self, S) 

self.__P = P 

def _repr_(self): 

""" 

Return a string representation of this scheme point. 

TESTS:: 

sage: from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal 

sage: P2.<x, y, z> = ProjectiveSpace(2, QQ) 

sage: pt = SchemeTopologicalPoint_prime_ideal(P2, y*z-x^2); pt 

Point on Projective Space of dimension 2 over Rational Field defined by the Ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field 

sage: pt._repr_() 

'Point on Projective Space of dimension 2 over Rational Field defined by the Ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field' 

""" 

return "Point on %s defined by the %s"%(self.scheme(), 

self.prime_ideal()) 

def prime_ideal(self): 

""" 

Return the prime ideal that defines this scheme point. 

EXAMPLES:: 

sage: from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal 

sage: P2.<x, y, z> = ProjectiveSpace(2, QQ) 

sage: pt = SchemeTopologicalPoint_prime_ideal(P2, y*z-x^2) 

sage: pt.prime_ideal() 

Ideal (-x^2 + y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field 

""" 

return self.__P 

def _richcmp_(self, other, op): 

""" 

Compare ``self`` to ``other``. 

TESTS:: 

sage: S = Spec(ZZ) 

sage: x = S(ZZ.ideal(5)) 

sage: y = S(ZZ.ideal(7)) 

sage: x == y 

False 

""" 

return richcmp(self.__P, other.__P, op) 

######################################################## 

# Points on a scheme defined by a morphism 

######################################################## 

def is_SchemeRationalPoint(x): 

return isinstance(x, SchemeRationalPoint) 

class SchemeRationalPoint(SchemePoint): 

def __init__(self, f): 

""" 

INPUT: 

- ``f`` - a morphism of schemes 

""" 

SchemePoint.__init__(self, f.codomain(), parent=f.parent()) 

self.__f = f 

def _repr_(self): 

return "Point on %s defined by the morphism %s"%(self.scheme(), 

self.morphism()) 

def morphism(self): 

return self.__f