Trac #34474: Make FiniteRankFreeModule.tensor_module(0, 1) return the…

… dual

In Sage 9.7.rc0, we have
{{{
sage: M = FiniteRankFreeModule(ZZ, 2, name='M')
sage: M.tensor_module(0, 1) is M.dual()
False
}}}
The identity of the two objects, which is mathematically exact, is
implemented here. This also simplifies some coercion maps.

URL: https://trac.sagemath.org/34474
Reported by: egourgoulhon
Ticket author(s): Eric Gourgoulhon
Reviewer(s): Matthias Koeppe

src/sage/tensor/modules/ext_pow_free_module.py

@@ -573,21 +573,12 @@ class ExtPowerDualFreeModule(FiniteRankFreeModule_abstract):
         sage: latex(M.dual())
         M^*
 
-    Since any tensor of type (0,1) is a linear form, there is a coercion map
-    from the set `T^{(0,1)}(M)` of such tensors to `M^*`::
+    It also coincides with the module of type-`(0,1)` tensors::
 
-        sage: T01 = M.tensor_module(0,1) ; T01
-        Free module of type-(0,1) tensors on the Rank-3 free module M over the
-         Integer Ring
-        sage: M.dual().has_coerce_map_from(T01)
-        True
-
-    There is also a coercion map in the reverse direction::
-
-        sage: T01.has_coerce_map_from(M.dual())
+        sage: M.dual_exterior_power(1) is M.tensor_module(0,1)
         True
 
-    For a degree `p\geq 2`, the coercion holds only in the direction
+    For a degree `p\geq 2`, there is a coercion map
     `\Lambda^p(M^*)\rightarrow T^{(0,p)}(M)`::
 
         sage: T02 = M.tensor_module(0,2) ; T02
@@ -598,24 +589,6 @@ class ExtPowerDualFreeModule(FiniteRankFreeModule_abstract):
         sage: A.has_coerce_map_from(T02)
         False
 
-    The coercion map `T^{(0,1)}(M) \rightarrow M^*` in action::
-
-        sage: b = T01([-2,1,4], basis=e, name='b') ; b
-        Type-(0,1) tensor b on the Rank-3 free module M over the Integer Ring
-        sage: b.display(e)
-        b = -2 e^0 + e^1 + 4 e^2
-        sage: lb = M.dual()(b) ; lb
-        Linear form b on the Rank-3 free module M over the Integer Ring
-        sage: lb.display(e)
-        b = -2 e^0 + e^1 + 4 e^2
-
-    The coercion map `M^* \rightarrow T^{(0,1)}(M)` in action::
-
-        sage: tlb = T01(lb) ; tlb
-        Type-(0,1) tensor b on the Rank-3 free module M over the Integer Ring
-        sage: tlb == b
-        True
-
     The coercion map `\Lambda^2(M^*)\rightarrow T^{(0,2)}(M)` in action::
 
         sage: ta = T02(a) ; ta
@@ -783,47 +756,6 @@ def _an_element_(self):
         resu.set_comp()[ind] = self._fmodule._ring.an_element()
         return resu
 
-    def _coerce_map_from_(self, other):
-        r"""
-        Determine whether coercion to ``self`` exists from other parent.
-
-        EXAMPLES:
-
-        Sets of type-`(0,1)` tensors coerce to ``self`` if the degree is 1::
-
-            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
-            sage: L1 = M.dual_exterior_power(1) ; L1
-            Dual of the Rank-3 free module M over the Integer Ring
-            sage: T01 = M.tensor_module(0,1) ; T01
-            Free module of type-(0,1) tensors on the Rank-3 free module M over
-             the Integer Ring
-            sage: L1._coerce_map_from_(T01)
-            True
-
-        Of course, coercions from other tensor types are meaningless::
-
-            sage: L1._coerce_map_from_(M.tensor_module(1,0))
-            False
-            sage: L1._coerce_map_from_(M.tensor_module(0,2))
-            False
-
-        If the degree is larger than 1, there is no coercion::
-
-            sage: L2 = M.dual_exterior_power(2) ; L2
-            2nd exterior power of the dual of the Rank-3 free module M over
-             the Integer Ring
-            sage: L2._coerce_map_from_(M.tensor_module(0,2))
-            False
-
-        """
-        from sage.tensor.modules.tensor_free_module import TensorFreeModule
-        if isinstance(other, TensorFreeModule):
-            # coercion of a type-(0,1) tensor to a linear form
-            if self._fmodule is other._fmodule and self._degree == 1 and \
-               other.tensor_type() == (0,1):
-                return True
-        return False
-
     #### End of parent methods
 
     @cached_method

src/sage/tensor/modules/finite_rank_free_module.py

@@ -1195,11 +1195,16 @@ def tensor_module(self, k, l):
             sage: M.tensor_module(1,2) is T
             True
 
-        The base module is itself the module of all type-`(1,0)` tensors::
+        The module of type-`(1,0)` tensors is the base module itself::
 
             sage: M.tensor_module(1,0) is M
             True
 
+        while the module of type-`(0,1)` tensors is the dual of the base module::
+
+            sage: M.tensor_module(0, 1) is M.dual()
+            True
+
         See :class:`~sage.tensor.modules.tensor_free_module.TensorFreeModule`
         for more documentation.
 
@@ -1209,6 +1214,8 @@ def tensor_module(self, k, l):
         except KeyError:
             if (k, l) == (1, 0):
                 T = self
+            elif (k, l) == (0, 1):
+                T = self.dual()
             else:
                 from sage.tensor.modules.tensor_free_module import TensorFreeModule
                 T = TensorFreeModule(self, (k,l))

src/sage/tensor/modules/tensor_free_module.py

@@ -237,39 +237,11 @@ class TensorFreeModule(FiniteRankFreeModule_abstract):
         sage: ta.symmetries() # the antisymmetry is of course preserved
         no symmetry;  antisymmetry: (0, 1)
 
-    For the degree `p=1`, there is a coercion in both directions::
+    For the degree `p=1`, we have the identity `\Lambda^1(M^*) = T^{(0,1)}(M) = M^*`::
 
-        sage: L1 = M.dual_exterior_power(1) ; L1
-        Dual of the Rank-3 free module M over the Integer Ring
-        sage: T01 = M.tensor_module(0,1) ; T01
-        Free module of type-(0,1) tensors on the Rank-3 free module M over the
-         Integer Ring
-        sage: T01.has_coerce_map_from(L1)
-        True
-        sage: L1.has_coerce_map_from(T01)
+        sage: M.dual_exterior_power(1) is M.tensor_module(0,1)
         True
-
-    The coercion map `\Lambda^1(M^*)\rightarrow T^{(0,1)}(M)` in action::
-
-        sage: a = M.linear_form('a')
-        sage: a[:] = -2, 4, 1 ; a.display(e)
-        a = -2 e^0 + 4 e^1 + e^2
-        sage: a.parent() is L1
-        True
-        sage: ta = T01(a) ; ta
-        Type-(0,1) tensor a on the Rank-3 free module M over the Integer Ring
-        sage: ta.display(e)
-        a = -2 e^0 + 4 e^1 + e^2
-
-    The coercion map `T^{(0,1)}(M) \rightarrow \Lambda^1(M^*)` in action::
-
-        sage: ta.parent() is T01
-        True
-        sage: lta = L1(ta) ; lta
-        Linear form a on the Rank-3 free module M over the Integer Ring
-        sage: lta.display(e)
-        a = -2 e^0 + 4 e^1 + e^2
-        sage: lta == a
+        sage: M.tensor_module(0,1) is M.dual()
         True
 
     There is a canonical identification between tensors of type `(1,1)` and
@@ -571,7 +543,7 @@ def _coerce_map_from_(self, other):
 
         but not to tensor modules of other types::
 
-            sage: M.tensor_module(0,1)._coerce_map_from_(End(M))
+            sage: M.tensor_module(0,2)._coerce_map_from_(End(M))
             False
 
         and not to type-`(1,1)` tensor modules defined on another free module::
@@ -597,8 +569,6 @@ def _coerce_map_from_(self, other):
 
         Coercion from alternating forms::
 
-            sage: M.tensor_module(0,1)._coerce_map_from_(M.dual_exterior_power(1))
-            True
             sage: M.tensor_module(0,2)._coerce_map_from_(M.dual_exterior_power(2))
             True
             sage: M.tensor_module(0,2)._coerce_map_from_(M.dual_exterior_power(3))
@@ -645,9 +615,6 @@ def _repr_(self):
             sage: M.tensor_module(1,1)
             Free module of type-(1,1) tensors on the 2-dimensional vector space
              M over the Rational Field
-            sage: M.tensor_module(0,1)
-            Free module of type-(0,1) tensors on the 2-dimensional vector space
-             M over the Rational Field
 
         """
         description = "Free module of type-({},{}) tensors on the {}".format(