Adding support for quotients of quotients.

Author: tscrim

src/sage/modules/free_module.py

@@ -1570,6 +1570,41 @@ def submodule(self, gens, check=True):
                                       "a submodule of self" % gens)
         return V
 
+    def quotient_module(self, sub, check=True):
+        r"""
+        Return the quotient of ``self`` by the given subspace ``sub``.
+
+        INPUT:
+
+        - ``sub`` -- a submodule of ``self`` or something that can
+          be turned into one via ``self.submodule(sub)``
+        - ``check`` -- (default: ``True``) whether or not to check that
+          ``sub`` is a submodule
+
+        EXAMPLES::
+
+            sage: S.<x,y,z> = PolynomialRing(QQ)
+            sage: M = S**2
+            sage: N = M.submodule([vector([x - y, z]), vector([y * z, x * z])])
+            sage: M.quotient(N)
+            Quotient module by Submodule of Ambient free module of rank 2 over
+             the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field
+            Generated by the rows of the matrix:
+            [x - y     z]
+            [  y*z   x*z]
+        """
+        if isinstance(sub, Module_free_ambient) and self.base_ring() != sub.base_ring():
+            raise ValueError("base rings must be the same")
+        if check and (not isinstance(sub, Module_free_ambient) or not sub.is_submodule(self)):
+            try:
+                sub = self.submodule(sub)
+            except (TypeError, ArithmeticError):
+                raise ArithmeticError("sub must be a subspace of self")
+        from .quotient_module import QuotientModule_free_ambient
+        return QuotientModule_free_ambient(self, sub)
+
+    quotient = quotient_module
+
     def __truediv__(self, sub):
         """
         Return the quotient of ``self`` by the given submodule sub.
@@ -4120,15 +4155,12 @@ def quotient(self, sub, check=True, **kwds):
 
         INPUT:
 
-        -  ``sub`` - a submodule of self, or something that can
-           be turned into one via self.submodule(sub).
-
-        -  ``check`` - (default: True) whether or not to check
-           that sub is a submodule.
-
-        -  further named arguments, that are passed to the constructor
-           of the quotient space.
-
+        - ``sub`` -- a submodule of ``self``, or something that can
+          be turned into one via ``self.submodule(sub)``
+        - ``check`` -- (default: ``True``) whether or not to check
+          that ``sub`` is a submodule
+        - further named arguments, that are passed to the constructor
+          of the quotient space
 
         EXAMPLES::
 
@@ -4146,8 +4178,8 @@ def quotient(self, sub, check=True, **kwds):
         if self.base_ring() == sage.rings.integer_ring.ZZ:
             from .fg_pid.fgp_module import FGP_Module
             return FGP_Module(self, sub, check=False, **kwds)
-        else:
-            raise NotImplementedError("quotients of modules over rings other than fields or ZZ is not fully implemented")
+
+        raise NotImplementedError("quotients of modules over rings other than fields or ZZ is not fully implemented")
 
 
 class FreeModule_generic_field(FreeModule_generic_pid):
@@ -5001,13 +5033,10 @@ def quotient(self, sub, check=True):
 
         INPUT:
 
-
-        -  ``sub`` - a submodule of self, or something that can
-           be turned into one via self.submodule(sub).
-
-        -  ``check`` - (default: True) whether or not to check
-           that sub is a submodule.
-
+        - ``sub`` -- a submodule of ``self``, or something that can
+          be turned into one via ``self.submodule(sub)``
+        - ``check`` -- (default: ``True``) whether or not to check
+          that ``sub`` is a submodule
 
         EXAMPLES::
 
@@ -6073,41 +6102,6 @@ def vector_space(self, base_field=None):
         else:
             return self.change_ring(base_field)
 
-    def quotient_module(self, sub, check=True):
-        """
-        Return the quotient of this free module by the given subspace ``sub``.
-
-        INPUT:
-
-        -  ``sub`` -- a submodule of this free module, or something that can
-           be turned into one via ``self.submodule(sub)``.
-
-        -  ``check`` -- (default: True) whether or not to check that ``sub`` is
-           a submodule
-
-        EXAMPLES::
-
-            sage: S.<x,y,z> = PolynomialRing(QQ)
-            sage: M = S**2
-            sage: N = M.submodule([vector([x - y, z]), vector([y * z, x * z])])
-            sage: M.quotient(N)
-            Quotient module by Submodule of Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field
-            Generated by the rows of the matrix:
-            [x - y     z]
-            [  y*z   x*z]
-        """
-        if isinstance(sub, Module_free_ambient) and self.base_ring() != sub.base_ring():
-            raise ValueError("base rings must be the same")
-        if check and (not isinstance(sub, Module_free_ambient) or not sub.is_submodule(self)):
-            try:
-                sub = self.submodule(sub)
-            except (TypeError, ArithmeticError):
-                raise ArithmeticError("sub must be a subspace of self")
-        from .quotient_module import QuotientModule_free_ambient
-        return QuotientModule_free_ambient(self, sub)
-
-    quotient = quotient_module
-
 
 ###############################################################################
 #

src/sage/modules/quotient_module.py

@@ -37,9 +37,8 @@ class QuotientModule_free_ambient(Module_free_ambient):
 
     INPUT:
 
-    - ``module`` -- an ambient free module
-
-    - ``sub`` -- a submodule of the ambient free module
+    - ``domain`` -- an ambient free module
+    - ``sub`` -- a submodule of ``domain``
 
     EXAMPLES::
 
@@ -68,9 +67,22 @@ def __init__(self, domain, sub):
         base_ring = domain.base_ring()
         degree = domain.degree()
         sparse = domain.is_sparse()
-        self._sub = sub
+        # We store these in order to retain the information of how this was constructed
         self._domain = domain
-        self.__hash = hash((domain, sub))
+        self._sub = sub
+        self.__hash = hash((self._domain, self._sub))
+
+        # We then convert the data into maps from free modules
+        if isinstance(domain, QuotientModule_free_ambient):
+            self._free_cover = domain.cover()
+            C = self._free_cover.element_class
+            v = [C(self._free_cover, x.list(), coerce=False, copy=False) for x in sub.gens()]
+            w = [C(self._free_cover, x.list(), coerce=False, copy=False) for x in domain.free_relations().gens()]
+            self._relations = self._free_cover.submodule(v + w, check=False)
+        else: # Otherwise domain should be a free module
+            self._free_cover = domain
+            self._relations = sub
+
         Module_free_ambient.__init__(self, base_ring, degree=degree, sparse=sparse)
 
     def _repr_(self):
@@ -161,6 +173,15 @@ def _coerce_map_from_(self, M):
         if isinstance(M, FreeModule_ambient):
             return (self.base_ring().has_coerce_map_from(M.base_ring()) and
                     self.degree() == M.degree())
+        from sage.modules.submodule import Subquotient_free_ambient
+        if isinstance(M, Subquotient_free_ambient):
+            return self._domain.has_coerce_map_from(self.ambient_module())
+        if (isinstance(M, QuotientModule_free_ambient)
+            and M.free_cover() == self.free_cover()):
+            try:
+                return M.free_relations().is_submodule(self.free_relations())
+            except NotImplementedError:
+                pass
 
     def ambient_module(self):
         """
@@ -196,7 +217,7 @@ def cover(self):
 
     def relations(self):
         r"""
-        Given this quotient space `Q = V/W`, return `W`.
+        Given this quotient space `Q = V/W`, return `W`
 
         EXAMPLES::
 
@@ -211,6 +232,66 @@ def relations(self):
 
     W = relations
 
+    def free_cover(self):
+        r"""
+        Given this quotient space `Q = V/W`, return the free module
+        that covers `V`.
+
+        EXAMPLES::
+
+            sage: S.<x,y,z> = PolynomialRing(QQ)
+            sage: M = S**2
+            sage: N = M.submodule([vector([x - y, z]), vector([y*z, x*z])])
+            sage: Q = M / N
+            sage: NQ = Q.submodule([Q([1,x])])
+            sage: QNQ = Q / NQ
+            sage: QNQ.free_cover() is Q.free_cover() is M
+            True
+
+        Note that this is different than the immediate cover::
+
+            sage: QNQ.cover() is Q
+            True
+            sage: QNQ.cover() is QNQ.free_cover()
+            False
+        """
+        return self._free_cover
+
+    V = cover
+
+    def free_relations(self):
+        r"""
+        Given this quotient space `Q = V/W`, return the submodule
+        that generates all relations of `Q`.
+
+        When `V` is a free module, then this returns `W`. Otherwise this
+        returns the union of `W` lifted to the cover of `V` and the relations
+        of `V` (repeated until `W` is a submodule of a free module).
+
+        EXAMPLES::
+
+            sage: S.<x,y,z> = PolynomialRing(QQ)
+            sage: M = S**2
+            sage: N = M.submodule([vector([x - y, z]), vector([y*z, x*z])])
+            sage: Q = M / N
+            sage: NQ = Q.submodule([Q([1,x])])
+            sage: QNQ = Q / NQ
+            sage: QNQ.free_relations()
+            Submodule of Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field
+            Generated by the rows of the matrix:
+            [    1     x]
+            [x - y     z]
+            [  y*z   x*z]
+
+        Note that this is different than the defining relations::
+
+            sage: QNQ.relations() is NQ
+            True
+            sage: QNQ.relations() == QNQ.free_relations()
+            False
+        """
+        return self._relations
+
 
 ###############################################################################
 #

src/sage/modules/submodule.py

@@ -114,13 +114,28 @@ def _repr_(self):
 
             sage: S.<x,y,z> = PolynomialRing(QQ)
             sage: M = S**2
-            sage: M.submodule([vector([x - y, z]), vector([y*z, x*z])])
+            sage: N = M.submodule([vector([x - y, z]), vector([y*z, x*z])])
+            sage: N
             Submodule of Ambient free module of rank 2 over the integral domain
             Multivariate Polynomial Ring in x, y, z over Rational Field
             Generated by the rows of the matrix:
             [x - y     z]
             [  y*z   x*z]
+
+            sage: Q = M.quotient_module(N)
+            sage: NQ = Q.submodule([Q([1,x])])
+            sage: NQ
+            Subquotient of Ambient free module of rank 2 over the integral domain Multivariate Polynomial Ring in x, y, z over Rational Field
+            Generated by the rows of the matrix:
+            [1 x]
+            With relations matrix:
+            [x - y     z]
+            [  y*z   x*z]
         """
+        if isinstance(self._ambient, QuotientModule_free_ambient):
+            return ("Subquotient of %s\n" % self.ambient_module().free_cover() +
+                    "Generated by the rows of the matrix:\n%s" % self.matrix() +
+                    "\nWith relations matrix:\n%s" % self._ambient.free_relations().matrix())
         return ("Submodule of %s\n" % self.ambient_module() +
                 "Generated by the rows of the matrix:\n%s" % self.matrix())