schemes.py

r"""
Schemes
"""
#*****************************************************************************
#  Copyright (C) 2005      David Kohel <kohel@maths.usyd.edu>
#                          William Stein <wstein@math.ucsd.edu>
#                2008-2012 Nicolas M. Thiery <nthiery at users.sf.net>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#                  http://www.gnu.org/licenses/
#******************************************************************************

from sage.categories.category import Category
from sage.categories.category_types import Category_over_base
from sage.categories.homsets import HomsetsCategory
from sage.categories.sets_cat import Sets

class Schemes(Category):
    """
    The category of all schemes.

    EXAMPLES::

        sage: Schemes()
        Category of schemes

    ``Schemes`` can also be used to construct the category of schemes
    over a given base::

        sage: Schemes(Spec(ZZ))
        Category of schemes over Integer Ring

        sage: Schemes(ZZ)
        Category of schemes over Integer Ring

    .. TODO::

        Make ``Schemes()`` a singleton category (and remove
        :class:`Schemes` from the workaround in
        :meth:`.category_types.Category_over_base._test_category_over_bases`).

        This is currently incompatible with the dispatching below.

    TESTS::

        sage: TestSuite(Schemes()).run()

    Check that Hom sets of schemes are in the correct category::

        sage: Schemes().Homsets().super_categories()
        [Category of homsets]
    """

    @staticmethod
    def __classcall_private__(cls, X = None):
        """
        Implement the dispatching ``Schemes(ZZ)`` -> ``Schemes_over_base``.

        EXAMPLES::

            sage: Schemes()
            Category of schemes

            sage: Schemes(Spec(ZZ))
            Category of schemes over Integer Ring

            sage: Schemes(ZZ)
            Category of schemes over Integer Ring
        """
        if X is not None:
            from sage.schemes.generic.scheme import is_Scheme
            if not is_Scheme(X):
                X = Schemes()(X)
            return Schemes_over_base(X)
        else:
            return super(Schemes, cls).__classcall__(cls)

    def super_categories(self):
        """
        EXAMPLES::

            sage: Schemes().super_categories()
            [Category of sets]
        """
        return [Sets()]

    def _call_(self, x):
        """
        Construct a scheme from the data in ``x``

        EXAMPLES:

        Let us first construct the category of schemes::

            sage: S = Schemes(); S
            Category of schemes

        We create a scheme from a ring::

            sage: X = S(ZZ); X                  # indirect doctest
            Spectrum of Integer Ring

        We create a scheme from a scheme (do nothing)::

            sage: S(X)
            Spectrum of Integer Ring

        We create a scheme morphism from a ring homomorphism.x::

            sage: phi = ZZ.hom(QQ); phi
            Ring Coercion morphism:
              From: Integer Ring
              To:   Rational Field
            sage: f = S(phi); f                 # indirect doctest
            Affine Scheme morphism:
              From: Spectrum of Rational Field
              To:   Spectrum of Integer Ring
              Defn: Ring Coercion morphism:
                      From: Integer Ring
                      To:   Rational Field

            sage: f.domain()
            Spectrum of Rational Field
            sage: f.codomain()
            Spectrum of Integer Ring
            sage: S(f)                          # indirect doctest
            Affine Scheme morphism:
              From: Spectrum of Rational Field
              To:   Spectrum of Integer Ring
              Defn: Ring Coercion morphism:
                      From: Integer Ring
                      To:   Rational Field

        """
        from sage.schemes.generic.scheme import is_Scheme
        if is_Scheme(x):
            return x
        from sage.schemes.generic.morphism import is_SchemeMorphism
        if is_SchemeMorphism(x):
            return x
        from sage.rings.morphism import is_RingHomomorphism
        from sage.rings.ring import CommutativeRing
        from sage.schemes.generic.spec import Spec
        if isinstance(x, CommutativeRing):
            return Spec(x)
        elif is_RingHomomorphism(x):
            A = Spec(x.codomain())
            return A.hom(x)
        else:
            raise TypeError("No way to create an object or morphism in %s from %s"%(self, x))


#############################################################
# Schemes over a given base scheme.
#############################################################
class Schemes_over_base(Category_over_base):
    """
    The category of schemes over a given base scheme.

    EXAMPLES::

        sage: Schemes(Spec(ZZ))
        Category of schemes over Integer Ring

    TESTS::

        sage: C = Schemes(ZZ)
        sage: TestSuite(C).run()
    """

    def base_scheme(self):
        """
        EXAMPLES::

            sage: Schemes(Spec(ZZ)).base_scheme()
            Spectrum of Integer Ring
        """
        return self.base()

    def super_categories(self):
        """
        EXAMPLES::

            sage: Schemes(Spec(ZZ)).super_categories()
            [Category of schemes]
        """
        return [Schemes()]

    def _repr_object_names(self):
        """
        EXAMPLES::

            sage: Schemes(Spec(ZZ)) # indirect doctest
            Category of schemes over Integer Ring
        """
        # To work around the name of the class (schemes_over_base)
        from sage.schemes.generic.scheme import is_AffineScheme
        if is_AffineScheme(self.base_scheme()):
            return "schemes over %s" % self.base_scheme().coordinate_ring()
        else:
            return "schemes over %s" % self.base_scheme()