def space(self):
r'''
Calculates the space of cocyles modulo coboundaries, as a Z-module.
TESTS:
sage: from darmonpoints.sarithgroup import *
sage: from darmonpoints.cohomology_abstract import *
sage: from darmonpoints.ocmodule import *
sage: GS = BigArithGroup(5, 6,1,use_shapiro=False,outfile='/tmp/darmonpoints.tmp') # optional - magma
sage: G = GS.large_group() # optional - magma
sage: V = OCVn(5,1) # optional - magma
sage: Coh = CohomologyGroup(G,V,trivial_action = False) # optional - magma
'''
verb = get_verbose()
set_verbose(0)
V = self.coefficient_module()
R = V.base_ring()
Vdim = V.dimension()
G = self.group()
gens = G.gens()
ambient = R**(Vdim * len(gens))
# Now find the subspace of cocycles
A = Matrix(R, Vdim * len(gens), 0)
for r in G.get_relation_words():
Alist = self.fox_gradient(r)
newA = block_matrix(Alist, nrows = 1)
A = A.augment(newA.transpose())
A = A.transpose()
cocycles = ambient.submodule([ambient(o) for o in A.right_kernel_matrix().rows()])
gmat = block_matrix([self._gen_pows[i][1] - 1 for i in range(len(G.gens()))], nrows = len(G.gens()))
coboundaries = cocycles.submodule([ambient(o) for o in gmat.columns()])
ans = cocycles.quotient(coboundaries)
set_verbose(verb)
return ans
def acting_matrix(self, g, prec=None):
dim = len(self.basis())
ans = Matrix(self._V.base_ring(), dim, 0)
for v in self.basis():
gv = g * v
gvlist = []
for w in gv._val:
try:
wlist = list(w)
except TypeError:
wlist = list(w._moments)
if prec is None:
gvlist.extend(wlist)
else:
gvlist.extend(wlist[:prec])
ans = ans.augment(Matrix(self._V.base_ring(), dim, 1, gvlist))
return ans
def acting_matrix(self, g, prec = None):
dim = len(self.basis())
ans = Matrix(self._V.base_ring(),dim, 0)
for v in self.basis():
gv = g * v
gvlist = []
for w in gv._val:
try:
wlist = list(w)
except TypeError:
wlist = list(w._moments)
if prec is None:
gvlist.extend(wlist)
else:
gvlist.extend(wlist[:prec])
ans = ans.augment(Matrix(self._V.base_ring(),dim,1,gvlist))
return ans
def space(self):
r'''
Calculates the space of cocyles modulo coboundaries, as a Z-module.
TESTS:
sage: from darmonpoints.sarithgroup import *
sage: from darmonpoints.cohomology_abstract import *
sage: from darmonpoints.ocmodule import *
sage: GS = BigArithGroup(5, 6,1,use_shapiro=False,outfile='/tmp/darmonpoints.tmp') # optional - magma
sage: G = GS.large_group() # optional - magma
sage: V = OCVn(5,1) # optional - magma
sage: Coh = CohomologyGroup(G,V,trivial_action = False) # optional - magma
'''
verb = get_verbose()
set_verbose(0)
V = self.coefficient_module()
R = V.base_ring()
Vdim = V.dimension()
G = self.group()
gens = G.gens()
ambient = R**(Vdim * len(gens))
# Now find the subspace of cocycles
A = Matrix(R, Vdim * len(gens), 0)
for nr, r in enumerate(G.get_relation_words()):
set_verbose(verb)
verbose('Processing relation word %s' % nr)
set_verbose(0)
Alist = [
MatrixSpace(R, Vdim, Vdim)(self.GA_to_local(o))
for o in self.fox_gradient(tuple(r))
]
newA = block_matrix(Alist, nrows=1)
A = A.augment(newA.transpose())
A = A.transpose()
cocycles = ambient.submodule(
[ambient(o) for o in A.right_kernel_matrix().rows()])
gmat = block_matrix(
[self._acting_matrix(g, Vdim) - 1 for g in G.gens()],
nrows=len(G.gens()))
coboundaries = cocycles.submodule([ambient(o) for o in gmat.columns()])
ans = cocycles.quotient(coboundaries)
set_verbose(verb)
return ans
