Trac #7946: Fix TestSuite failures for schemes

Consider the following situation:
{{{
sage: S = Spec(ZZ)
sage: x = S.an_element()
}}}
Running `TestSuite(S)` and `TestSuite(x)` yields several failures.  A
related problem is
{{{
sage: S
Spectrum of Integer Ring
sage: parent(x)
Set of rational points of Spectrum of Integer Ring
}}}
whereas we expect `parent(x) is S`.

Here is the complete `TestSuite` report:
{{{
sage: TestSuite(S).run()
Failure in _test_an_element:
Traceback (most recent call last):
...
NotImplementedError
------------------------------------------------------------
  Failure in _test_category:
  Traceback (most recent call last):
  ...
  AssertionError: False is not true
  ------------------------------------------------------------
  Failure in _test_pickling:
  Traceback (most recent call last):
  ...
  AssertionError: Point on Spectrum of Integer Ring defined by the
Principal ideal (991) of Integer Ring != Point on Spectrum of Integer
Ring defined by the Principal ideal (991) of Integer Ring
  ------------------------------------------------------------
  The following tests failed: _test_category, _test_pickling
Failure in _test_elements
Failure in _test_some_elements:
Traceback (most recent call last):
...
NotImplementedError
------------------------------------------------------------
The following tests failed: _test_an_element, _test_elements,
_test_some_elements
}}}
{{{
sage: TestSuite(x).run()
Failure in _test_category:
Traceback (most recent call last):
...
AssertionError: False is not true
------------------------------------------------------------
Failure in _test_pickling:
Traceback (most recent call last):
...
AssertionError: Point on Spectrum of Integer Ring defined by the
Principal ideal (991) of Integer Ring != Point on Spectrum of Integer
Ring defined by the Principal ideal (991) of Integer Ring
------------------------------------------------------------
The following tests failed: _test_category, _test_pickling
}}}

(Note: the `NotImplementedError` that one gets after applying #16158 is
a different one than before.)

URL: http://trac.sagemath.org/7946
Reported by: nthiery
Ticket author(s): Peter Bruin
Reviewer(s): Travis Scrimshaw

src/sage/schemes/generic/morphism.py

@@ -782,7 +782,10 @@ def _call_(self, x):
 
         The image scheme point.
 
-        EXAMPLES::
+        EXAMPLES:
+
+        The following fails because inverse images of prime ideals
+        under ring homomorphisms are not yet implemented::
 
             sage: R.<x> = PolynomialRing(QQ)
             sage: phi = R.hom([QQ(7)])
@@ -791,10 +794,7 @@ def _call_(self, x):
             sage: f(X.an_element())    # indirect doctest
             Traceback (most recent call last):
             ...
-            TypeError: Point on Spectrum of Rational Field defined by the
-            Principal ideal (0) of Rational Field fails to convert into the
-            map's domain Spectrum of Rational Field, but a `pushforward`
-            method is not properly implemented
+            NotImplementedError
         """
         # By virtue of argument preprocessing in __call__, we can assume that
         # x is a topological scheme point of self

src/sage/schemes/generic/point.py

@@ -20,12 +20,13 @@ class SchemePoint(Element):
     Base class for points on a scheme, either topological or defined
     by a morphism.
     """
-    def __init__(self, S):
+    def __init__(self, S, parent=None):
         """
         INPUT:
 
+        - ``S`` -- a scheme
 
-        -  ``S`` - a scheme
+        - ``parent`` -- the parent in which to construct this point
 
         TESTS::
 
@@ -34,7 +35,7 @@ def __init__(self, S):
             sage: P = SchemePoint(S); P
             Point on Spectrum of Integer Ring
         """
-        Element.__init__(self, S.Hom(S))
+        Element.__init__(self, parent)
         self.__S = S
 
     def scheme(self):
@@ -74,19 +75,40 @@ def is_SchemeTopologicalPoint(x):
     return isinstance(x, SchemeTopologicalPoint)
 
 class SchemeTopologicalPoint(SchemePoint):
-    pass
+    """
+    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`
 
-        -  ``u`` - morphism with domain U an affine scheme
-
-        -  ``x`` - point on U
+        - ``x`` -- topological point on `U`
         """
-        SchemePoint.__init__(self, u.codomain())
+        SchemeTopologicalPoint.__init__(self, u.codomain())
         self.__u = u
         self.__x = x
 
@@ -147,8 +169,10 @@ def __init__(self, S, P, check=False):
         """
         R = S.coordinate_ring()
         from sage.rings.ideal import is_Ideal
-        if not (is_Ideal(P) and P.ring() is R):
+        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
@@ -186,6 +210,20 @@ def prime_ideal(self):
         """
         return self.__P
 
+    def __cmp__(self, other):
+        """
+        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 cmp(self.__P, other.__P)
+
 ########################################################
 # Points on a scheme defined by a morphism
 ########################################################
@@ -201,7 +239,7 @@ def __init__(self, f):
 
         -  ``f`` - a morphism of schemes
         """
-        SchemePoint.__init__(self, f.codomain())
+        SchemePoint.__init__(self, f.codomain(), parent=f.parent())
         self.__f = f
 
     def _repr_(self):

src/sage/schemes/generic/scheme.py

@@ -21,17 +21,19 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-
 from sage.structure.parent import Parent
 from sage.misc.all import cached_method
 from sage.rings.all import (IntegerRing,
                             ZZ, GF, PowerSeriesRing,
                             Rationals)
 
 from sage.rings.commutative_ring import is_CommutativeRing
+from sage.rings.ideal import is_Ideal
 from sage.rings.morphism import is_RingHomomorphism
 from sage.structure.unique_representation import UniqueRepresentation
 
+from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal
+
 def is_Scheme(x):
     """
     Test whether ``x`` is a scheme.
@@ -93,14 +95,16 @@ def __init__(self, X=None, category=None):
         """
         Construct a scheme.
 
-        TESTS::
+        TESTS:
+
+        The full test suite works since :trac:`7946`::
 
             sage: R.<x, y> = QQ[]
             sage: I = (x^2 - y^2)*R
             sage: RmodI = R.quotient(I)
             sage: X = Spec(RmodI)
-            sage: TestSuite(X).run(skip = ["_test_an_element", "_test_elements",
-            ...                            "_test_some_elements", "_test_category"]) # See #7946
+            sage: TestSuite(X).run()
+
         """
         from sage.schemes.generic.morphism import is_SchemeMorphism
 
@@ -162,7 +166,7 @@ def _morphism(self, *args, **kwds):
 
         TESTS:
 
-        This shows that issue at trac ticket 7389 is solved::
+        This shows that issue at :trac:`7389` is solved::
 
             sage: S = Spec(ZZ)
             sage: f = S.identity_morphism()
@@ -912,7 +916,7 @@ def __call__(self, *args):
             sage: P = S(ZZ.ideal(3)); P
             Point on Spectrum of Integer Ring defined by the Principal ideal (3) of Integer Ring
             sage: type(P)
-            <class 'sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal'>
+            <class 'sage.schemes.generic.point.AffineScheme_with_category.element_class'>
             sage: S(ZZ.ideal(next_prime(1000000)))
             Point on Spectrum of Integer Ring defined by the Principal ideal (1000003) of Integer Ring
 
@@ -936,15 +940,69 @@ def __call__(self, *args):
             Set of morphisms
               From: Spectrum of Integer Ring
               To:   Spectrum of Integer Ring
+
+        For affine or projective varieties, passing the correct number
+        of elements of the base ring constructs the rational point
+        with these elements as coordinates::
+
+            sage: S = AffineSpace(ZZ, 1)
+            sage: S(0)
+            (0)
+
+        To prevent confusion with this usage, topological points must
+        be constructed by explicitly specifying a prime ideal, not
+        just generators::
+
+            sage: R = S.coordinate_ring()
+            sage: S(R.ideal(0))
+            Point on Affine Space of dimension 1 over Integer Ring defined by the Ideal (0) of Multivariate Polynomial Ring in x over Integer Ring
+
+        This explains why the following example raises an error rather
+        than constructing the topological point defined by the prime
+        ideal `(0)` as one might expect::
+
+            sage: S = Spec(ZZ)
+            sage: S(0)
+            Traceback (most recent call last):
+            ...
+            TypeError: cannot call Spectrum of Integer Ring with arguments (0,)
         """
         if len(args) == 1:
-            from sage.rings.ideal import is_Ideal
             x = args[0]
-            if is_Ideal(x) and x.ring() is self.coordinate_ring():
-                from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal
-                return SchemeTopologicalPoint_prime_ideal(self, x)
+            if ((isinstance(x, self.element_class) and (x.parent() is self or x.parent() == self))
+                or (is_Ideal(x) and x.ring() is self.coordinate_ring())):
+                # Construct a topological point from x.
+                return self._element_constructor_(x)
+        try:
+            # Construct a scheme homset or a scheme-valued point from
+            # args using the generic Scheme.__call__() method.
+            return super(AffineScheme, self).__call__(*args)
+        except NotImplementedError:
+            # This arises from self._morphism() not being implemented.
+            # We must convert it into a TypeError to keep the coercion
+            # system working.
+            raise TypeError('cannot call %s with arguments %s' % (self, args))
+
+    Element = SchemeTopologicalPoint_prime_ideal
 
-        return super(AffineScheme, self).__call__(*args)
+    def _element_constructor_(self, x):
+        """
+        Construct a topological point from `x`.
+
+        TESTS::
+
+            sage: S = Spec(ZZ)
+            sage: S(ZZ.ideal(0))
+            Point on Spectrum of Integer Ring defined by the Principal ideal (0) of Integer Ring
+        """
+        if isinstance(x, self.element_class):
+            if x.parent() is self:
+                return x
+            elif x.parent() == self:
+                return self.element_class(self, x.prime_ideal())
+        elif is_Ideal(x) and x.ring() is self.coordinate_ring():
+            return self.element_class(self, x)
+        raise TypeError('cannot convert %s to a topological point of %s' % (x, self))
 
     def _an_element_(self):
         r"""

src/sage/schemes/generic/spec.py

@@ -43,7 +43,7 @@ def Spec(R, S=None):
         Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
         sage: X = Spec(PolynomialRing(GF(49,'a'), 3, 'x')); X
         Spectrum of Multivariate Polynomial Ring in x0, x1, x2 over Finite Field in a of size 7^2
-        sage: TestSuite(X).run(skip=["_test_an_element", "_test_elements", "_test_some_elements"])
+        sage: TestSuite(X).run()
 
     Applying ``Spec`` twice to the same ring gives identical output
     (see :trac:`17008`)::