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Standardizing submodules as subquotients.
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@tscrim
tscrim committed Jun 23, 2022
1 parent 9bd6f89 commit 568097b
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207 changes: 121 additions & 86 deletions src/sage/modules/free_module.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -745,7 +745,7 @@ def basis_seq(V, vecs):
"""
for z in vecs:
z.set_immutable()
return Sequence(vecs, universe=V, check = False, immutable=True, cr=True)
return Sequence(vecs, universe=V, check=False, immutable=True, cr=True)


###############################################################################
Expand Down Expand Up @@ -916,30 +916,6 @@ def is_sparse(self):
"""
return self.__is_sparse

def ambient_module(self):
"""
Return the ambient module associated to this module.
EXAMPLES::
sage: R.<x,y> = QQ[]
sage: M = FreeModule(R,2)
sage: M.ambient_module()
Ambient free module of rank 2 over the integral domain
Multivariate Polynomial Ring in x, y over Rational Field
::
sage: V = FreeModule(QQ, 4).span([[1,2,3,4], [1,0,0,0]]); V
Vector space of degree 4 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 0 0]
[ 0 1 3/2 2]
sage: V.ambient_module()
Vector space of dimension 4 over Rational Field
"""
return FreeModule(self.base_ring(), self.degree())

def _an_element_(self):
"""
Return an arbitrary element of a free module.
Expand All @@ -959,7 +935,7 @@ def _an_element_(self):
try:
return self.gen(0)
except ValueError:
return self(0)
return self.zero()

def some_elements(self):
r"""
Expand Down Expand Up @@ -1084,6 +1060,36 @@ def zero_submodule(self):
"""
return self.submodule([], check=False)

def relations_matrix(self):
r"""
Return the matrix of relations of ``self``.
EXAMPLES::
sage: V = GF(2)^2
sage: V.relations_matrix()
[]
sage: W = V.subspace([[1, 0]])
sage: W.relations_matrix()
[]
sage: Q = V / W
sage: Q.relations_matrix()
[1 0]
sage: S.<x,y,z> = PolynomialRing(QQ)
sage: M = S**2
sage: M.relations_matrix()
[]
sage: N = M.submodule([vector([x - y, z]), vector([y*z, x*z])])
sage: Q = M.quotient_module(N)
sage: Q.relations_matrix()
[x - y z]
[ y*z x*z]
"""
return self.relations().matrix()

def __richcmp__(self, other, op):
r"""
Rich comparison via containment in the same ambient space.
Expand Down Expand Up @@ -1370,6 +1376,18 @@ def _eq(self, other):
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.quotient_module(N)
sage: M._eq(Q)
False
sage: M.zero_submodule()._eq(Q.zero_submodule())
False
sage: Q.zero_submodule()._eq(M.zero_submodule())
False
sage: M.zero_submodule()._eq(N.zero_submodule())
True
"""
if self.degree() != other.degree():
return False
Expand All @@ -1392,7 +1410,7 @@ def _eq(self, other):
rx = other.zero_submodule()
if lx != rx:
return False
# NOTE: This method should be overwritten for free modules.
# NOTE: This method is overwritten for free modules.
# Subsequently, we know we are not an ambient free module!

# self and other are not ambient.
Expand Down Expand Up @@ -1440,32 +1458,12 @@ def is_submodule(self, other):
if self.base_ring() != other.base_ring():
return False

from sage.modules.submodule import Submodule_free_ambient
if isinstance(self, Submodule_free_ambient):
if isinstance(other, Submodule_free_ambient):
if self.defining_module() != other.defining_module():
return False
if not (self.defining_module() == other.defining_module()):
raise NotImplementedError("could not determine containment")
else:
if self.defining_module() != other:
return False
if not (self.defining_module() == other):
raise NotImplementedError("could not determine containment")
elif isinstance(other, Submodule_free_ambient):
if other.defining_module() != self:
return False
if not (other.defining_module() == self):
if not (self.ambient_module() == other.ambient_module()):
if not (self.ambient_module() != other.ambient_module()):
raise NotImplementedError("could not determine containment")

try:
if self.ambient_vector_space() != other.ambient_vector_space():
return False
if other is other.ambient_vector_space():
return True
except AttributeError:
# Not all modules have an ambient_vector_space.
pass
return False
if other is self.ambient_module():
return True

from sage.modules.quotient_module import QuotientModule_free_ambient
if isinstance(other, QuotientModule_free_ambient):
Expand Down Expand Up @@ -1527,23 +1525,20 @@ def span(self, gens, base_ring=None, check=True, already_echelonized=False):
if isinstance(gens, FreeModule_generic):
gens = gens.gens()
if base_ring is None or base_ring is self.base_ring():
from .submodule import Submodule_free_ambient
if isinstance(self, Submodule_free_ambient):
ambient = self.defining_module()
else:
ambient = self.ambient_module()
return Submodule_free_ambient(ambient, gens, check=check)
else:
try:
M = self.change_ring(base_ring)
except TypeError:
raise ValueError("Argument base_ring (= %s) is not compatible " % base_ring +
"with the base ring (= %s)." % self.base_ring())
try:
return M.span(gens)
except TypeError:
raise ValueError("Argument gens (= %s) is not compatible " % gens +
"with base_ring (= %s)." % base_ring)
from sage.modules.submodule import Subquotient_free_ambient
return Subquotient_free_ambient(self.ambient_module(), gens, check=check)

# The base ring has changed
try:
M = self.change_ring(base_ring)
except TypeError:
raise ValueError("argument base_ring (= %s) is not compatible " % base_ring +
"with the base ring (= %s)" % self.base_ring())
try:
return M.span(gens)
except TypeError:
raise ValueError("argument gens (= %s) is not compatible " % gens +
"with base_ring (= %s)" % base_ring)

def submodule(self, gens, check=True):
r"""
Expand Down Expand Up @@ -1575,6 +1570,21 @@ def submodule(self, gens, check=True):
"a submodule of self" % gens)
return V

def __truediv__(self, sub):
"""
Return the quotient of ``self`` by the given submodule sub.
EXAMPLES::
sage: V1 = ZZ^2; W1 = V1.span([[1,2],[3,4]])
sage: V1/W1
Finitely generated module V/W over Integer Ring with invariants (2)
sage: V2 = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W2 = V2.span([2*V2.0+4*V2.1, 9*V2.0+12*V2.1, 4*V2.2])
sage: V2/W2
Finitely generated module V/W over Integer Ring with invariants (4, 12)
"""
return self.quotient(sub, check=True)


class FreeModule_generic(Module_free_ambient):
"""
Expand Down Expand Up @@ -3252,6 +3262,26 @@ def _mul_(self, other, switch_sides=False):
B = other * B if switch_sides else B * other
return self.span(B.rows())

# Subclasses should override this method when appropriate
def relations(self):
"""
Return the module of relations of ``self``.
EXAMPLES::
sage: V = GF(2)^2
sage: V.relations() == V.zero_submodule()
True
sage: W = V.subspace([[1, 0]])
sage: W.relations() == V.zero_submodule()
True
sage: Q = V / W
sage: Q.relations() == W
True
"""
return self.zero_submodule()


class FreeModule_generic_domain(FreeModule_generic):
"""
Expand Down Expand Up @@ -4119,21 +4149,6 @@ def quotient(self, sub, check=True, **kwds):
else:
raise NotImplementedError("quotients of modules over rings other than fields or ZZ is not fully implemented")

def __truediv__(self, sub):
"""
Return the quotient of ``self`` by the given submodule sub.
EXAMPLES::
sage: V1 = ZZ^2; W1 = V1.span([[1,2],[3,4]])
sage: V1/W1
Finitely generated module V/W over Integer Ring with invariants (2)
sage: V2 = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W2 = V2.span([2*V2.0+4*V2.1, 9*V2.0+12*V2.1, 4*V2.2])
sage: V2/W2
Finitely generated module V/W over Integer Ring with invariants (4, 12)
"""
return self.quotient(sub, check=True)


class FreeModule_generic_field(FreeModule_generic_pid):
"""
Expand Down Expand Up @@ -5268,7 +5283,7 @@ def _coerce_map_from_(self, M):
sage: V = QQ^2
sage: V.coerce_map_from(M)
"""
from sage.modules.submodule import Submodule_free_ambient
from sage.modules.submodule import Subquotient_free_ambient
from sage.modules.quotient_module import FreeModule_ambient_field_quotient

if isinstance(M, FreeModule_ambient_field_quotient):
Expand All @@ -5281,7 +5296,7 @@ def _coerce_map_from_(self, M):
# complexity of this is quadratic in space and time,
# since it constructs a matrix.
return True
elif isinstance(M, Submodule_free_ambient):
elif isinstance(M, Subquotient_free_ambient):
if (self.base_ring().has_coerce_map_from(M.base_ring()) and
self.rank() == M.degree()):
return True
Expand Down Expand Up @@ -6723,6 +6738,26 @@ def ambient_module(self):
"""
return self.__ambient_module

def relations(self):
r"""
Return the submodule defining the relations of ``self`` as a
subquotient (considering the ambient module as a quotient module).
EXAMPLES::
sage: V = GF(2)^2
sage: W = V.subspace([[1, 0]])
sage: W.relations() == V.zero_submodule()
True
sage: Q = V / W
sage: Q.relations() == W
True
sage: Q.zero_submodule().relations() == W
True
"""
return self.__ambient_module.relations()

def echelon_coordinates(self, v, check=True):
r"""
Write `v` in terms of the echelonized basis for ``self``.
Expand Down
24 changes: 6 additions & 18 deletions src/sage/modules/quotient_module.py
View file
Original file line number Diff line number Diff line change
Expand Up @@ -162,32 +162,20 @@ def _coerce_map_from_(self, M):
return (self.base_ring().has_coerce_map_from(M.base_ring()) and
self.degree() == M.degree())

def submodule(self, gens, check=True):
r"""
Create the `R`-submodule of ``self`` with given generators,
where `R` is the base ring of ``self``.
INPUT:
- ``gens`` -- a list of free module elements or a free module
- ``check`` -- (default: ``True``) whether or not to verify
that the gens are in ``self``
def ambient_module(self):
"""
Return ``self``, since ``self`` is ambient.
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.quotient_module(N)
sage: Q.submodule([])
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:
[]
sage: Q.ambient_module() is Q
True
"""
if isinstance(gens, Module_free_ambient):
gens = gens.gens()
from .submodule import Submodule_free_ambient
return Submodule_free_ambient(self, gens, check=check)
return self

def cover(self):
r"""
Expand Down
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