Implementing SchemeMorphism_sum

src/sage/categories/homset.py

@@ -116,6 +116,8 @@
          from Alternating group of order 3!/2 as a permutation group
            to Alternating group of order 3!/2 as a permutation group
            in Category of finite enumerated permutation groups
+        sage: Hom(G, G) is End(G)
+        True
         sage: Hom(ZZ, QQ, Sets())
         Set of Morphisms from Integer Ring to Rational Field in Category of sets
 

src/sage/matrix/action.pyx

@@ -408,10 +408,9 @@ cdef class MatrixPolymapAction(MatrixMulAction):
             sage: H = Hom(P,P)
             sage: A = MatrixPolymapAction(M,H)
             sage: A
-            Left action by Full MatrixSpace of 2 by 2 dense matrices over Rational
-            Field on Set of morphisms
-              From: Projective Space of dimension 1 over Rational Field
-              To:   Projective Space of dimension 1 over Rational Field
+            Left action by Full MatrixSpace of 2 by 2 dense matrices over
+             Rational Field on Set of scheme endomorphisms of Projective Space of
+             dimension 1 over Rational Field
         """
         if not isinstance(S, SchemeHomset_generic):
             raise TypeError("not a scheme polynomial morphism: %s"% S)
@@ -427,9 +426,7 @@ cdef class MatrixPolymapAction(MatrixMulAction):
             sage: H = End(P)
             sage: A = MatrixPolymapAction(M,H)
             sage: A.codomain()
-            Set of morphisms
-              From: Projective Space of dimension 1 over Rational Field
-              To:   Projective Space of dimension 1 over Rational Field
+            Set of scheme endomorphisms of Projective Space of dimension 1 over Rational Field
             sage: A.codomain().is_endomorphism_set()
             True
         """
@@ -479,10 +476,9 @@ cdef class PolymapMatrixAction(MatrixMulAction):
             sage: H = Hom(P,P)
             sage: A = PolymapMatrixAction(M,H)
             sage: A
-            Right action by Full MatrixSpace of 2 by 2 dense matrices over Rational
-            Field on Set of morphisms
-              From: Projective Space of dimension 1 over Rational Field
-              To:   Projective Space of dimension 1 over Rational Field
+            Right action by Full MatrixSpace of 2 by 2 dense matrices over
+             Rational Field on Set of scheme endomorphisms of Projective Space of
+             dimension 1 over Rational Field
         """
         if not isinstance(S, SchemeHomset_generic):
             raise TypeError("not a scheme polynomial morphism: %s"% S)
@@ -500,9 +496,7 @@ cdef class PolymapMatrixAction(MatrixMulAction):
             sage: H = End(P)
             sage: A = PolymapMatrixAction(M,H)
             sage: A.codomain()
-            Set of morphisms
-              From: Projective Space of dimension 1 over Rational Field
-              To:   Projective Space of dimension 1 over Rational Field
+            Set of scheme endomorphisms of Projective Space of dimension 1 over Rational Field
             sage: A.codomain().is_endomorphism_set()
             True
         """

src/sage/rings/ring_extension.pyx

@@ -669,6 +669,7 @@ cdef class RingExtension_generic(CommutativeRing):
             sage: dir(K)                                                                # needs sage.rings.number_field
             ['CartesianProduct',
              'Element',
+             'End',
              'Hom',
              ...
              'zeta',

src/sage/schemes/affine/affine_homset.py

@@ -113,9 +113,7 @@ class SchemeHomset_polynomial_affine_space(SchemeHomset_generic):
 
         sage: A.<x,y> = AffineSpace(2, QQ)
         sage: Hom(A, A)
-        Set of morphisms
-          From: Affine Space of dimension 2 over Rational Field
-          To:   Affine Space of dimension 2 over Rational Field
+        Set of scheme endomorphisms of Affine Space of dimension 2 over Rational Field
     """
     def identity(self):
         """
@@ -126,9 +124,7 @@ def identity(self):
             sage: A.<x,y> = AffineSpace(2, QQ)
             sage: I = A.identity_morphism()
             sage: I.parent()
-            Set of morphisms
-              From: Affine Space of dimension 2 over Rational Field
-              To:   Affine Space of dimension 2 over Rational Field
+            Set of scheme endomorphisms of Affine Space of dimension 2 over Rational Field
             sage: _.identity() == I
             True
         """

src/sage/schemes/affine/affine_morphism.py

@@ -273,6 +273,19 @@ def __call__(self, x, check=True):
             P = [f(x._coords) for f in self._polys]
         return self.codomain().point_homset(R)(P, check=check)
 
+    def __hash__(self):
+        """
+        Return a hash for this scheme morphism.
+
+        EXAMPLES::
+
+            sage: A.<x,y> = AffineSpace(QQ, 2)
+            sage: F = A.hom([2*x, 2*y], A)
+            sage: hash(F) == hash((F.__class__, A, A, (2*x, 2*y)))
+            True
+        """
+        return hash((self.__class__, self.domain(), self.codomain(), self._polys))
+
     def __eq__(self, right):
         """
         Tests the equality of two affine maps.

src/sage/schemes/affine/affine_space.py

@@ -371,9 +371,7 @@ def _homset(self, *args, **kwds):
 
             sage: A.<x,y> = AffineSpace(2, QQ)
             sage: Hom(A, A)
-            Set of morphisms
-              From: Affine Space of dimension 2 over Rational Field
-              To:   Affine Space of dimension 2 over Rational Field
+            Set of scheme endomorphisms of Affine Space of dimension 2 over Rational Field
         """
         return SchemeHomset_polynomial_affine_space(*args, **kwds)
 

src/sage/schemes/curves/projective_curve.py

@@ -220,6 +220,41 @@
 
         Curve_generic.__init__(self, A, X, category=category)
 
+    def _an_element_(self):
+        r"""
+        Return an element of this projective curve.
+
+        ALGORITHM:
+
+        We try to find variables that never appear as a pure monomial, and fail otherwise.
+
+        EXAMPLES::
+
+            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
+            sage: C = Curve(x*y^2*z^7 - x^10 - x^2*z^8)
+            sage: C.an_element()
+            (0 : 1 : 0)
+            sage: C.an_element() in C
+            True
+        """
+        # The dimension won't be too large anyways.
+        gens = list(self.ambient_space().gens())
+        counts = [0] * len(gens)
+        for poly in self.defining_polynomials():
+            for m in poly.monomials():
+                mvs = m.variables()
+                if len(mvs) == 1:
+                    counts[gens.index(mvs[0])] += 1
+        if min(counts) > 0:
+            return NotImplemented
+
+        # Set this to 1 and others to 0
+        ring = self.base_ring()
+        idx = counts.index(0)
+        coordinate = [ring.zero()] * len(gens)
+        coordinate[idx] = ring.one()
+        return self(coordinate)
+
     def _repr_type(self):
         r"""
         Return a string representation of the type of this curve.

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -3166,7 +3166,7 @@ def montgomery_model(self, twisted=False, morphism=False):
             sage: P + Q             # this doesn't work...
             Traceback (most recent call last):
             ...
-            TypeError: unsupported operand parent(s) for +: ...
+            NotImplementedError
             sage: f(g(P) + g(Q))    # ...but this does
             (107 : 168 : 1)
 

src/sage/schemes/elliptic_curves/hom_sum.py

@@ -60,14 +60,22 @@
 
 class EllipticCurveHom_sum(EllipticCurveHom):
 
-    _degree = None
     _phis = None
+    _degree = None
 
     def __init__(self, phis, domain=None, codomain=None):
         r"""
         Construct a sum morphism of elliptic curves from its summands.
         (For empty sums, the domain and codomain curves must be given.)
 
+        .. TODO::
+
+            This class :class:`EllipticCurveHom_sum` really should inherit from
+            :class:`SchemeMorphism_sum`. However, there's a lot of issues that
+            I can't figure out how to resolve, mainly regarding
+            multi-inheritance in Python and which methods the coercion model
+            should use. See :issue:`37705` for some of my thoughts.
+
         EXAMPLES::
 
             sage: from sage.schemes.elliptic_curves.hom_sum import EllipticCurveHom_sum

src/sage/schemes/elliptic_curves/homset.py

@@ -57,7 +57,6 @@
 # ****************************************************************************
 
 from sage.rings.integer_ring import ZZ
-from sage.categories.morphism import Morphism
 from sage.schemes.generic.homset import SchemeHomset_generic
 from sage.schemes.elliptic_curves.ell_generic import EllipticCurve_generic
 
@@ -231,24 +230,6 @@ def _repr_(self):
         s += f'\n  To:   {self.codomain()}'
         return s
 
-    def identity(self):
-        r"""
-        Return the identity morphism in this elliptic-curve homset
-        as an :class:`EllipticCurveHom` object.
-
-        EXAMPLES::
-
-            sage: E = EllipticCurve(j=42)
-            sage: End(E).identity()
-            Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + 5901*x + 1105454 over Rational Field
-              Via:  (u,r,s,t) = (1, 0, 0, 0)
-            sage: End(E).identity() == E.scalar_multiplication(1)
-            True
-        """
-        if not self.is_endomorphism_set():
-            raise ValueError('domain and codomain must be equal')
-        return self.domain().identity_morphism()
-
     def is_commutative(self):
         r"""
         Assuming this homset is an endomorphism ring, check whether

src/sage/schemes/generic/homset.py

@@ -176,7 +176,7 @@
             from sage.categories.schemes import Schemes
             category = Schemes(base_spec)
         key = (id(X), id(Y), category, as_point_homset)
-        extra = {'X':X, 'Y':Y, 'base_ring':base_ring, 'check':check}
+        extra = {'X': X, 'Y': Y, 'base_ring': base_ring, 'check': check}
         return key, extra
 
     def create_object(self, version, key, **extra_args):
@@ -245,9 +245,7 @@
         sage: from sage.schemes.generic.homset import SchemeHomset_generic
         sage: A2 = AffineSpace(QQ,2)
         sage: Hom = SchemeHomset_generic(A2, A2); Hom
-        Set of morphisms
-          From: Affine Space of dimension 2 over Rational Field
-          To:   Affine Space of dimension 2 over Rational Field
+        Set of scheme endomorphisms of Affine Space of dimension 2 over Rational Field
         sage: Hom.category()
         Category of endsets of schemes over Rational Field
     """
@@ -265,8 +263,9 @@
             sage: loads(Hom.dumps()) == Hom
             True
         """
-        return SchemeHomset, (self.domain(), self.codomain(), self.homset_category(),
-                              self.base_ring(), False, False)
+        return SchemeHomset, (self.domain(), self.codomain(),
+                              self.homset_category(), self.base_ring(), False,
+                              False)
 
     def __call__(self, *args, **kwds):
         r"""
@@ -295,15 +294,20 @@
         EXAMPLES::
 
             sage: A = AffineSpace(4, QQ)
-            sage: print(A.structure_morphism()._repr_())
-            Scheme morphism:
+            sage: print(repr(A.structure_morphism().parent()))
+            Set of morphisms
               From: Affine Space of dimension 4 over Rational Field
               To:   Spectrum of Rational Field
-              Defn: Structure map
+
+            sage: print(repr(Spec(QQ).End()))
+            Set of scheme endomorphisms of Spectrum of Rational Field
         """
-        s = 'Set of morphisms'
-        s += '\n  From: %s' % self.domain()
-        s += '\n  To:   %s' % self.codomain()
+        if self.is_endomorphism_set():
+            return f"Set of scheme endomorphisms of {self.domain()}"
+
+        s = "Set of morphisms"
+        s += f'\n  From: {self.domain()}'
+        s += f'\n  To:   {self.codomain()}'
         return s
 
     def natural_map(self):
@@ -324,13 +328,74 @@
               From: Affine Space of dimension 4 over Rational Field
               To:   Spectrum of Rational Field
               Defn: Structure map
+
+            sage: Spec(QQ).End().natural_map()
+            Scheme endomorphism of Spectrum of Rational Field
+              Defn: Identity map
         """
         X = self.domain()
         Y = self.codomain()
         if is_AffineScheme(Y) and Y.coordinate_ring() == X.base_ring():
             return SchemeMorphism_structure_map(self)
+        if self.is_endomorphism_set():
+            return self.identity()
         raise NotImplementedError
 
+    def identity(self):
+        r"""
+        Return the identity morphism in this homset.
+
+        Only implemented when this homset is an endomorphism set. In this case,
+        either the :meth:`identity_morphism` of the domain is returned, or a
+        :class:`SchemeMorphism_id` class is constructed.
+
+        .. SEEALSO:: :meth:`natural_map`
+
+        EXAMPLES::
+
+            sage: End(Spec(QQ)).identity()
+            Scheme endomorphism of Spectrum of Rational Field
+              Defn: Identity map
+
+            sage: Hom(Spec(ZZ), Spec(QQ)).identity()
+            Traceback (most recent call last):
+            ...
+            ValueError: domain and codomain must be equal
+
+        Elliptic curves have a custom :meth:`identity_morphism`::
+
+            sage: E = EllipticCurve(j=42)
+            sage: End(E).identity()
+            Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + 5901*x + 1105454 over Rational Field
+              Via:  (u,r,s,t) = (1, 0, 0, 0)
+            sage: End(E).identity() == E.scalar_multiplication(1)
+            True
+        """
+        if not self.is_endomorphism_set():
+            raise ValueError('domain and codomain must be equal')
+        try:
+            return self.domain().identity_morphism()
+        except AttributeError:
+            from sage.schemes.generic.morphism import SchemeMorphism_id
+            return SchemeMorphism_id(self.domain())
+
+    def _an_element_(self):
+        r"""
+        Return a morphism from this homset via the
+        :meth:`natural_map`, or :meth:`zero` if a natural map does
+        not exist.
+
+        EXAMPLES::
+
+            sage: Spec(QQ).End().an_element()
+            Scheme endomorphism of Spectrum of Rational Field
+              Defn: Identity map
+        """
+        try:
+            return self.natural_map()
+        except NotImplementedError:
+            return self.zero()
+
     def _element_constructor_(self, x, check=True):
         """
         Construct a scheme morphism.
@@ -380,11 +445,9 @@
             sage: A.<x,y> = AffineSpace(R)
             sage: C = A.subscheme(x*y - 1)
             sage: H = C.Hom(C); H
-            Set of morphisms
-              From: Closed subscheme of Affine Space of dimension 2 over Rational Field
-                    defined by: x*y - 1
-              To:   Closed subscheme of Affine Space of dimension 2 over Rational Field
-                    defined by: x*y - 1
+            Set of scheme endomorphisms of Closed subscheme of Affine Space of
+             dimension 2 over Rational Field defined by:
+             x*y - 1
             sage: H(1)
             Traceback (most recent call last):
             ...
@@ -401,7 +464,6 @@
 
         raise TypeError("x must be a ring homomorphism, list or tuple")
 
-
 # *******************************************************************
 #  Base class for points
 # *******************************************************************
@@ -582,7 +644,7 @@
             except AttributeError:  # no .ambient_space
                 return False
         elif isinstance(other, SchemeHomset_points):
-        #we are converting between scheme points
+            #we are converting between scheme points
             source = other.codomain()
             if isinstance(target, AlgebraicScheme_subscheme):
                 #subscheme coerce when there is containment