point.py

"""
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

########################################################
# 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):
        """
        INPUT:
        
        
        -  ``S`` - a scheme

        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, S.Hom(S))
        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):
    pass

class SchemeTopologicalPoint_affine_open(SchemeTopologicalPoint):
    def __init__(self, u, x):
        """
        INPUT:
        
        
        -  ``u`` - morphism with domain U an affine scheme
        
        -  ``x`` - point on U
        """
        SchemePoint.__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

        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 Ideal
        P = Ideal(R, P)
        # 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

########################################################
# 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())
        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