def cohomology(self, d):
r"""
Return the ``d``-th cohomology group.
EXAMPLES::
sage: E.<x,y> = ExteriorAlgebra(QQ)
sage: H = E.hochschild_complex(E)
sage: H.cohomology(0)
Vector space of dimension 3 over Rational Field
sage: H.cohomology(1)
Vector space of dimension 4 over Rational Field
sage: H.cohomology(2)
Vector space of dimension 6 over Rational Field
sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: T = SGA.trivial_representation()
sage: H = SGA.hochschild_complex(T)
sage: H.cohomology(0)
Vector space of dimension 1 over Rational Field
sage: H.cohomology(1)
Vector space of dimension 0 over Rational Field
sage: H.cohomology(2)
Vector space of dimension 0 over Rational Field
When working over general rings (except `\ZZ`) and we can
construct a unitriangular basis for the image quotient,
we fallback to a slower implementation using (combinatorial)
free modules::
sage: R.<x,y> = QQ[]
sage: SGA = SymmetricGroupAlgebra(R, 2)
sage: T = SGA.trivial_representation()
sage: H = SGA.hochschild_complex(T)
sage: H.cohomology(1)
Free module generated by {} over Multivariate Polynomial Ring in x, y over Rational Field
"""
if self._A.category() is not self._A.category().FiniteDimensional():
raise NotImplementedError("the algebra must be finite dimensional")
maps = {
d + 1: self.coboundary(d + 1).matrix(),
d: self.coboundary(d).matrix()
}
C = ChainComplex(maps, degree_of_differential=1)
try:
return C.homology(d + 1)
except NotImplementedError:
pass
# Fallback if we are not working over a field or \ZZ
cb = self.coboundary(d)
cb1 = self.coboundary(d + 1)
ker = cb1.kernel()
im_retract = ker.submodule([ker.retract(b) for b in cb.image_basis()],
unitriangular=True)
return ker.quotient_module(im_retract)
def cohomology(self, d):
"""
Return the ``d``-th cohomology group.
EXAMPLES::
sage: E.<x,y> = ExteriorAlgebra(QQ)
sage: H = E.hochschild_complex(E)
sage: H.cohomology(0)
Vector space of dimension 3 over Rational Field
sage: H.cohomology(1)
Vector space of dimension 4 over Rational Field
sage: H.cohomology(2)
Vector space of dimension 6 over Rational Field
sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: T = SGA.trivial_representation()
sage: H = SGA.hochschild_complex(T)
sage: H.cohomology(0)
Vector space of dimension 1 over Rational Field
sage: H.cohomology(1)
Vector space of dimension 0 over Rational Field
sage: H.cohomology(2)
Vector space of dimension 0 over Rational Field
When working over general rings (except `\ZZ`) and we can
construct a unitriangular basis for the image quotient,
we fallback to a slower implementation using (combinatorial)
free modules::
sage: R.<x,y> = QQ[]
sage: SGA = SymmetricGroupAlgebra(R, 2)
sage: T = SGA.trivial_representation()
sage: H = SGA.hochschild_complex(T)
sage: H.cohomology(1)
Free module generated by {} over Multivariate Polynomial Ring in x, y over Rational Field
"""
if self._A.category() is not self._A.category().FiniteDimensional():
raise NotImplementedError("the algebra must be finite dimensional")
maps = {d+1: self.coboundary(d+1).matrix(), d: self.coboundary(d).matrix()}
C = ChainComplex(maps, degree_of_differential=1)
try:
return C.homology(d+1)
except NotImplementedError:
pass
# Fallback if we are not working over a field or \ZZ
cb = self.coboundary(d)
cb1 = self.coboundary(d+1)
ker = cb1.kernel()
im_retract = ker.submodule([ker.retract(b) for b in cb.image_basis()],
unitriangular=True)
return ker.quotient_module(im_retract)
def random_chain_complex(level=1):
"""
Return a random chain complex, defined by specifying a single
random matrix in a random degree, with differential of degree
either 1 or -1. The matrix is randomly sparse or dense.
:param level: measure of complexity: the larger this is, the
larger the matrix can be, and the larger its degree can be in
the chain complex.
:type level: positive integer; optional, default 1
EXAMPLES::
sage: from sage.homology.tests import random_chain_complex
sage: C = random_chain_complex()
sage: C
Chain complex with at most 2 nonzero terms over Integer Ring
sage: C.degree_of_differential() # random: either 1 or -1
1
"""
bound = 50*level
nrows = randint(0, bound)
ncols = randint(0, bound)
sparseness = bool(randint(0, 1))
mat = random_matrix(ZZ, nrows, ncols, sparse=sparseness)
dim = randint(-bound, bound)
deg = 2 * randint(0, 1) - 1 # -1 or 1
return ChainComplex({dim: mat}, degree = deg)
def SChainComplex(kchaincomplex, start=0, end=15):
r"""
Convert the KenzoChainComplex ``kchcm`` (between dimensions ``start`` and
``end``) to a ChainComplex.
INPUT:
- ``kchaincomplex``- A KenzoChainComplex
- ``start``- An integer number (optional, default 0)
- ``end``- An integer number greater than or equal to ``start`` (optional, default 15)
OUTPUT:
- A ChainComplex
EXAMPLES::
sage: from sage.interfaces.kenzo import KChainComplex, SChainComplex # optional - kenzo
sage: m1 = matrix(ZZ, 3, 2, [-1, 1, 3, -4, 5, 6])
sage: m4 = matrix(ZZ, 2, 2, [1, 2, 3, 6])
sage: m5 = matrix(ZZ, 2, 3, [2, 2, 2, -1, -1, -1])
sage: sage_chcm = ChainComplex({1: m1, 4: m4, 5: m5}, degree = -1) # optional - kenzo
sage: SChainComplex(KChainComplex(sage_chcm)) == sage_chcm # optional - kenzo
True
"""
matrices = {}
for i in range(start, end):
dffr_i = chcm_mat2(kchaincomplex._kenzo, i)
if ((nlig(dffr_i).python() != 0) and (ncol(dffr_i).python() != 0)):
matrices[i] = k2s_matrix(convertmatrice(dffr_i))
return ChainComplex(matrices, degree=-1)
def godement_cochain_complex(self):
"""
Construct the Godement cochain complex of ``self``.
"""
# The case that the domain_space has dimension 0
if self._domain_poset.height() == 1:
rank = sum(self._stalk_dict[key] for key in self._stalk_dict)
differential = matrix(self._base_ring, 0, rank)
return ChainComplex([rank, differential], base_ring=self._base_ring)
# Other cases
end_base = sorted([[x] for x in self._domain_poset.list()])
diff_dict = dict()
for p in range(1, self._domain_poset.height()):
start_base = end_base
end_base = list(filter(lambda c: len(c)==p+1, self._domain_poset.chains()))
end_base = sorted(end_base)
diff_dict[p-1] = self._godement_complex_differential(start_base, end_base)
return ChainComplex(diff_dict, base_ring = self._base_ring)
def cohomology_complex(self, m):
r"""
Return the "cohomology complex" `C^*(m)`
See [Klyachko]_, equation 4.2.
INPUT:
- ``m`` -- tuple of integers or `M`-lattice point. A point in
the dual lattice of the fan. Must be immutable.
OUTPUT:
The "cohomology complex" as a chain complex over the
:meth:`base_ring`.
EXAMPLES::
sage: P3 = toric_varieties.P(3)
sage: rays = [(1,0,0), (0,1,0), (0,0,1)]
sage: F1 = FilteredVectorSpace(rays, {0:[0], 1:[2], 2:[1]})
sage: F2 = FilteredVectorSpace(rays, {0:[1,2], 1:[0]})
sage: r = P3.fan().rays()
sage: V = P3.sheaves.Klyachko({r[0]:F1, r[1]:F2, r[2]:F2, r[3]:F2})
sage: tau = Cone([(1,0,0), (0,1,0)])
sage: sigma = Cone([(1, 0, 0)])
sage: M = P3.fan().dual_lattice()
sage: m = M(1, 1, 0); m.set_immutable()
sage: V.cohomology_complex(m)
Chain complex with at most 2 nonzero terms over Rational Field
sage: F = CyclotomicField(3)
sage: P3 = toric_varieties.P(3).change_ring(F)
sage: V = P3.sheaves.Klyachko({r[0]:F1, r[1]:F2, r[2]:F2, r[3]:F2})
sage: V.cohomology_complex(m)
Chain complex with at most 2 nonzero terms over Cyclotomic
Field of order 3 and degree 2
"""
fan = self._variety.fan()
C = fan.complex()
CV = []
F = self.base_ring()
for dim in range(1, fan.dim() + 1):
codim = fan.dim() - dim
d_C = C.differential(codim)
d_V = []
for j in range(d_C.ncols()):
tau = fan(dim)[j]
d_V_row = []
for i in range(d_C.nrows()):
sigma = fan(dim - 1)[i]
if sigma.is_face_of(tau):
pr = self.E_quotient_projection(sigma, tau, m)
d = d_C[i, j] * pr.matrix().transpose()
else:
E_sigma = self.E_quotient(sigma, m)
E_tau = self.E_quotient(tau, m)
d = zero_matrix(F, E_tau.dimension(),
E_sigma.dimension())
d_V_row.append(d)
d_V.append(d_V_row)
d_V = block_matrix(d_V, ring=F)
CV.append(d_V)
from sage.homology.chain_complex import ChainComplex
return ChainComplex(CV, base_ring=self.base_ring())
