r"""
Chain complexes
AUTHORS:
- John H. Palmieri (2009-04)
This module implements chain complexes of free `R`-modules, for any
commutative ring `R` (although the interesting things, like homology,
only work if `R` is the integers or a field).
Fix a ring `R`. A chain complex over `R` is a collection of
`R`-modules `\{C_n\}` indexed by the integers, with `R`-module maps
`d_n : C_n \rightarrow C_{n+1}` such that `d_{n+1} \circ d_n = 0` for
all `n`. The maps `d_n` are called *differentials*.
One can vary this somewhat: the differentials may decrease degree by
one instead of increasing it: sometimes a chain complex is defined
with `d_n : C_n \rightarrow C_{n-1}` for each `n`. Indeed, the
differentials may change dimension by any fixed integer.
Also, the modules may be indexed over an abelian group other than the
integers, e.g., `\ZZ^{m}` for some integer `m \geq 1`, in which
case the differentials may change the grading by any element of that
grading group.
In this implementation, the ring `R` must be commutative and the
modules `C_n` must be free `R`-modules. As noted above, homology
calculations will only work if the ring `R` is either `\ZZ` or a field.
The modules may be indexed by any free abelian group. The
differentials may increase degree by 1 or decrease it, or indeed
change it by any fixed amount: this is controlled by the ``degree``
parameter used in defining the chain complex.
"""
from sage.structure.sage_object import SageObject
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.modules.free_module import FreeModule, VectorSpace
from sage.matrix.matrix0 import Matrix
from sage.matrix.constructor import matrix, prepare_dict
from sage.misc.latex import latex
from sage.groups.additive_abelian.additive_abelian_group import AdditiveAbelianGroup_fixed_gens
from sage.modules.free_module import FreeModule
from sage.rings.all import GF, prime_range
def dhsw_snf(mat, verbose=False):
"""
Preprocess a matrix using the "Elimination algorithm" described by
Dumas et al., and then call ``elementary_divisors`` on the
resulting (smaller) matrix.
'dhsw' stands for 'Dumas, Heckenbach, Saunders, Welker,' and 'snf'
stands for 'Smith Normal Form.'
INPUT:
- ``mat`` - an integer matrix, either sparse or dense.
(They use the transpose of the matrix considered here, so they use
rows instead of columns.)
Algorithm: go through ``mat`` one column at a time. For each
column, add multiples of previous columns to it until either
- it's zero, in which case it should be deleted.
- its first nonzero entry is 1 or -1, in which case it should be kept.
- its first nonzero entry is something else, in which case it is
deferred until the second pass.
Then do a second pass on the deferred columns.
At this point, the columns with 1 or -1 in the first entry
contribute to the rank of the matrix, and these can be counted and
then deleted (after using the 1 or -1 entry to clear out its row).
Suppose that there were `N` of these.
The resulting matrix should be much smaller; we then feed it
to Sage's ``elementary_divisors`` function, and prepend `N` 1's to
account for the rows deleted in the previous step.
EXAMPLES::
sage: from sage.homology.chain_complex import dhsw_snf
sage: mat = matrix(ZZ, 3, 4, range(12))
sage: dhsw_snf(mat)
[1, 4, 0]
sage: mat = random_matrix(ZZ, 20, 20, x=-1, y=2)
sage: mat.elementary_divisors() == dhsw_snf(mat)
True
REFERENCES:
- Dumas, Heckenbach, Saunders, Welker, "Computing simplicial
homology based on efficient Smith normal form algorithms," in
"Algebra, geometry, and software systems" (2003), 177-206.
"""
ring = mat.base_ring()
rows = mat.nrows()
cols = mat.ncols()
new_data = {}
new_mat = matrix(ring, rows, cols, new_data)
add_to_rank = 0
zero_cols = 0
if verbose:
print "old matrix: %s by %s" % (rows, cols)
leading_positions = {}
if verbose:
print "starting pass 1"
for j in range(cols):
new_col = mat.matrix_from_columns([j])
if new_col.is_zero():
zero_cols += 1
else:
check_leading = True
while check_leading:
i = new_col.nonzero_positions_in_column(0)[0]
entry = new_col[i,0]
check_leading = False
if i in leading_positions:
for c in leading_positions[i]:
earlier = new_mat[i,c]
if entry and earlier.divides(entry):
quo = entry.divide_knowing_divisible_by(earlier)
new_col = new_col - quo * new_mat.matrix_from_columns([c])
entry = 0
if not new_col.is_zero():
check_leading = True
if not new_col.is_zero():
new_mat.set_column(j-zero_cols, new_col.column(0))
i = new_col.nonzero_positions_in_column(0)[0]
if i in leading_positions:
leading_positions[i].append(j-zero_cols)
else:
leading_positions[i] = [j-zero_cols]
else:
zero_cols += 1
cols = cols - zero_cols
zero_cols = 0
new_mat = new_mat.matrix_from_columns(range(cols))
if verbose:
print "starting pass 2"
keep_columns = range(cols)
check_leading = True
while check_leading:
check_leading = False
new_leading = leading_positions.copy()
for i in leading_positions:
if len(leading_positions[i]) > 1:
j = leading_positions[i][0]
jth = new_mat[i, j]
for n in leading_positions[i][1:]:
nth = new_mat[i,n]
if jth.divides(nth):
quo = nth.divide_knowing_divisible_by(jth)
new_mat.add_multiple_of_column(n, j, -quo)
elif nth.divides(jth):
quo = jth.divide_knowing_divisible_by(nth)
jth = nth
new_mat.swap_columns(n, j)
new_mat.add_multiple_of_column(n, j, -quo)
else:
(g,r,s) = jth.xgcd(nth)
(unit,A,B) = r.xgcd(-s)
jth_col = new_mat.column(j)
nth_col = new_mat.column(n)
new_mat.set_column(j, r*jth_col + s*nth_col)
new_mat.set_column(n, B*jth_col + A*nth_col)
nth = B*jth + A*nth
jth = g
quo = nth.divide_knowing_divisible_by(jth)
new_mat.add_multiple_of_column(n, j, -quo)
new_leading[i].remove(n)
if new_mat.column(n).is_zero():
keep_columns.remove(n)
zero_cols += 1
else:
new_r = new_mat.column(n).nonzero_positions()[0]
if new_r in new_leading:
new_leading[new_r].append(n)
else:
new_leading[new_r] = [n]
check_leading = True
leading_positions = new_leading
if verbose:
print "starting pass 3"
max_leading = 1
for i in leading_positions:
j = leading_positions[i][0]
entry = new_mat[i,j]
if entry.abs() == 1:
add_to_rank += 1
keep_columns.remove(j)
for c in new_mat.nonzero_positions_in_row(i):
if c in keep_columns:
new_mat.add_multiple_of_column(c, j, -entry * new_mat[i,c])
else:
max_leading = max(max_leading, new_mat[i,j].abs())
if max_leading != 1:
new_mat = new_mat.matrix_from_columns(keep_columns)
if verbose:
print "new matrix: %s by %s" % (new_mat.nrows(), new_mat.ncols())
if new_mat.is_sparse():
ed = [1]*add_to_rank + new_mat.dense_matrix().elementary_divisors()
else:
ed = [1]*add_to_rank + new_mat.elementary_divisors()
else:
if verbose:
print "new matrix: all pivots are 1 or -1"
ed = [1]*add_to_rank
if len(ed) < rows:
return ed + [0]*(rows - len(ed))
else:
return ed[:rows]
def _latex_module(R, m):
"""
LaTeX string representing a free module over ``R`` of rank ``m``.
INPUT:
- ``R`` - a commutative ring
- ``m`` - non-negative integer
This is used by the ``_latex_`` method for chain complexes.
EXAMPLES::
sage: from sage.homology.chain_complex import _latex_module
sage: _latex_module(ZZ, 3)
'\\Bold{Z}^{3}'
sage: _latex_module(ZZ, 0)
'0'
sage: _latex_module(GF(3), 1)
'\\Bold{F}_{3}^{1}'
"""
if m == 0:
return str(latex(0))
else:
return str(latex(FreeModule(R, m)))
class ChainComplex(SageObject):
"""
Define a chain complex.
INPUT:
- ``data`` - the data defining the chain complex; see below for
more details.
- ``base_ring`` - a commutative ring (optional), the ring over
which the chain complex is defined. If this is not specified,
it is determined by the data defining the chain complex.
- ``grading_group`` - a free abelian group (optional, default
ZZ), the group over which the chain complex is indexed.
- ``degree`` - element of grading_group (optional, default 1),
the degree of the differential.
- ``check_products`` - boolean (optional, default True). If True,
check that each consecutive pair of differentials are
composable and have composite equal to zero.
OUTPUT: a chain complex
.. warning::
Right now, homology calculations will only work if the base
ring is either ZZ or a field, so please take this into account
when defining a chain complex.
Use data to define the chain complex. This may be in any of the
following forms.
1. a dictionary with integers (or more generally, elements of
grading_group) for keys, and with data[n] a matrix representing
(via left multiplication) the differential coming from degree
`n`. (Note that the shape of the matrix then determines the
rank of the free modules `C_n` and `C_{n+d}`.)
2. a list or tuple of the form `[C_0, d_0, C_1, d_1, C_2, d_2,
...]`, where each `C_i` is a free module and each `d_i` is a
matrix, as above. This only makes sense if ``grading_group``
is `\\ZZ` and ``degree`` is 1.
3. a list or tuple of the form `[r_0, d_0, r_1, d_1, r_2, d_2,
...]`, where `r_i` is the rank of the free module `C_i` and
each `d_i` is a matrix, as above. This only makes sense if
``grading_group`` is `\\ZZ` and ``degree`` is 1.
4. a list or tuple of the form `[d_0, d_1, d_2, ...]` where each
`d_i` is a matrix, as above. This only makes sense if
``grading_group`` is `\\ZZ` and ``degree`` is 1.
.. note::
In fact, the free modules `C_i` in case 2 and the ranks `r_i`
in case 3 are ignored: only the matrices are kept, and from
their shapes, the ranks of the modules are determined.
(Indeed, if ``data`` is a list or tuple, then any element which
is not a matrix is discarded; thus the list may have any number
of different things in it, and all of the non-matrices will be
ignored.) No error checking is done to make sure, for
instance, that the given modules have the appropriate ranks for
the given matrices. However, as long as ``check_products`` is
True, the code checks to see if the matrices are composable and
that each appropriate composite is zero.
If the base ring is not specified, then the matrices are examined
to determine a ring over which they are all naturally defined, and
this becomes the base ring for the complex. If no such ring can
be found, an error is raised. If the base ring is specified, then
the matrices are converted automatically to this ring when
defining the chain complex. If some matrix cannot be converted,
then an error is raised.
EXAMPLES::
sage: ChainComplex()
Chain complex with at most 0 nonzero terms over Integer Ring
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C
Chain complex with at most 2 nonzero terms over Integer Ring
sage: D = ChainComplex([matrix(ZZ, 2, 2, [0, 1, 0, 0]), matrix(ZZ, 2, 2, [0, 1, 0, 0])], base_ring=GF(2)); D
Chain complex with at most 3 nonzero terms over Finite Field of size 2
sage: D == loads(dumps(D))
True
Note that when a chain complex is defined in Sage, new
differentials may be created: every nonzero module in the chain
complex must have a differential coming from it, even if that
differential is zero::
sage: IZ = ChainComplex({0: identity_matrix(ZZ, 1)})
sage: IZ.differential() # the differentials in the chain complex
{0: [1], 1: []}
sage: IZ.differential(1).parent()
Full MatrixSpace of 0 by 1 dense matrices over Integer Ring
sage: mat = ChainComplex({0: matrix(ZZ, 3, 4)}).differential(1)
sage: mat.nrows(), mat.ncols()
(0, 3)
Defining the base ring implicitly::
sage: ChainComplex([matrix(QQ, 3, 1), matrix(ZZ, 4, 3)])
Chain complex with at most 3 nonzero terms over Rational Field
sage: ChainComplex([matrix(GF(125, 'a'), 3, 1), matrix(ZZ, 4, 3)])
Chain complex with at most 3 nonzero terms over Finite Field in a of size 5^3
If the matrices are defined over incompatible rings, an error results::
sage: ChainComplex([matrix(GF(125, 'a'), 3, 1), matrix(QQ, 4, 3)])
Traceback (most recent call last):
...
TypeError: unable to find a common ring for all elements
If the base ring is given explicitly but is not compatible with
the matrices, an error results::
sage: ChainComplex([matrix(GF(125, 'a'), 3, 1)], base_ring=QQ)
Traceback (most recent call last):
...
TypeError: Unable to coerce 0 (<type 'sage.rings.finite_rings.element_givaro.FiniteField_givaroElement'>) to Rational
"""
def __init__(self, data=None, **kwds):
"""
See ``ChainComplex`` for full documentation.
EXAMPLES::
sage: C = ChainComplex(); C
Chain complex with at most 0 nonzero terms over Integer Ring
sage: D = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: D
Chain complex with at most 2 nonzero terms over Integer Ring
"""
check_products = kwds.get('check_products', True) or kwds.get('check_diffs')
base_ring = kwds.get('base_ring', None)
grading_group = kwds.get('grading_group', ZZ)
degree = kwds.get('degree', 1)
try:
deg = grading_group(degree)
except:
raise ValueError, "The 'degree' does not appear to be an element of the grading group."
new_data = {}
if data is None or (isinstance(data, (list, tuple)) and len(data) == 0):
try:
zero = grading_group.zero_vector()
except AttributeError:
zero = ZZ.zero_element()
if base_ring is None:
base_ring = ZZ
new_data = {zero: matrix(base_ring, 0, 0, [])}
else:
if isinstance(data, dict):
temp_dict = data
elif isinstance(data, (list, tuple)):
if degree != 1:
raise ValueError, "The degree must be +1 if the data argument is a list or tuple."
if grading_group != ZZ:
raise ValueError, "The grading_group must be ZZ if the data argument is a list or tuple."
new_list = filter(lambda x: isinstance(x, Matrix), data)
temp_dict = dict(zip(range(len(new_list)), new_list))
else:
raise TypeError, "The data for a chain complex must be a dictionary, list, or tuple."
if base_ring is None:
junk, ring = prepare_dict(dict([n, temp_dict[n].base_ring()(0)] for n in temp_dict))
base_ring = ring
for n in temp_dict.keys():
if not n in grading_group:
raise ValueError, "One of the dictionary keys does not appear to be an element of the grading group."
mat = temp_dict[n]
if not isinstance(mat, Matrix):
raise TypeError, "One of the differentials in the data dictionary indexed by does not appear to be a matrix."
if mat.base_ring() == base_ring:
new_data[n] = mat
else:
new_data[n] = mat.change_ring(base_ring)
for n in temp_dict.keys():
mat = temp_dict[n]
if n+degree in temp_dict:
mat2 = temp_dict[n+degree]
if check_products:
try:
prod = mat2 * mat
except TypeError:
raise TypeError, "The differentials d_{%s} and d_{%s} are not compatible: their product is not defined." % (n, n+degree)
if not prod.is_zero():
raise ValueError, "The differentials d_{%s} and d_{%s} are not compatible: their composition is not zero." % (n, n+degree)
else:
if not mat.nrows() == 0:
if n+2*degree in temp_dict:
new_data[n+degree] = matrix(base_ring, temp_dict[n+2*degree].ncols(), mat.nrows())
else:
new_data[n+degree] = matrix(base_ring, 0, mat.nrows())
self._grading_group = grading_group
self._degree = deg
self._base_ring = base_ring
self._diff = new_data
self._ranks = {}
def base_ring(self):
"""
The base ring for this simplicial complex.
EXAMPLES::
sage: ChainComplex().base_ring()
Integer Ring
"""
return self._base_ring
def differential(self, dim=None):
"""
The differentials which make up the chain complex.
INPUT:
- ``dim`` - element of the grading group (optional, default
None). If this is None, return a dictionary of all of the
differentials. If this is a single element, return the
differential starting in that dimension.
OUTPUT: either a dictionary of all of the differentials or a single
differential (i.e., a matrix)
EXAMPLES::
sage: D = ChainComplex({0: matrix(ZZ, 2, 2, [1,0,0,2])})
sage: D.differential()
{0: [1 0]
[0 2], 1: []}
sage: D.differential(0)
[1 0]
[0 2]
sage: C = ChainComplex({0: identity_matrix(ZZ, 40)})
sage: C.differential()
{0: 40 x 40 dense matrix over Integer Ring, 1: []}
"""
if dim is None:
return self._diff
deg = self._degree
if dim in self._diff:
return self._diff[dim]
if dim+deg in self._diff:
return matrix(self._base_ring, self._diff[dim+deg].ncols(), 0)
return matrix(self._base_ring, 0, 0)
def dual(self):
"""
The dual chain complex to ``self``.
Since all modules in ``self`` are free of finite rank, the
dual in dimension `n` is isomorphic to the original chain
complex in dimension `n`, and the corresponding boundary
matrix is the transpose of the matrix in the original complex.
This converts a chain complex to a cochain complex and vice
versa.
EXAMPLES::
sage: C = ChainComplex({2: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C._degree
1
sage: C.differential(2)
[3 0 0]
[0 0 0]
sage: C.dual()._degree
-1
sage: C.dual().differential(3)
[3 0]
[0 0]
[0 0]
"""
data = {}
deg = self._degree
for d in self.differential():
data[(d+deg)] = self.differential()[d].transpose()
return ChainComplex(data, degree=-deg)
def free_module(self):
"""
The free module underlying this chain complex.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0]), 1: matrix(ZZ, 0, 2)})
sage: C.free_module()
Ambient free module of rank 5 over the principal ideal domain Integer Ring
This defines the forgetful functor from the category of chain
complexes to the category of free modules::
sage: FreeModules(ZZ)(C)
Ambient free module of rank 5 over the principal ideal domain Integer Ring
"""
rank = sum([mat.ncols() for mat in self.differential().values()])
return FreeModule(self._base_ring, rank)
def __cmp__(self, other):
"""
Return True iff this chain complex is the same as other: that
is, if the base rings and the matrices of the two are the
same.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])}, base_ring=GF(2))
sage: D = ChainComplex({0: matrix(GF(2), 2, 3, [1, 0, 0, 0, 0, 0]), 1: matrix(ZZ, 0, 2), 3: matrix(ZZ, 0, 0)}) # base_ring determined from the matrices
sage: C == D
True
"""
if self._base_ring != other._base_ring:
return -1
R = self._base_ring
equal = True
for d,mat in self.differential().iteritems():
if d not in other.differential():
equal = equal and mat.ncols() == 0 and mat.nrows() == 0
else:
equal = (equal and
other.differential()[d].change_ring(R) == mat.change_ring(R))
for d,mat in other.differential().iteritems():
if d not in self.differential():
equal = equal and mat.ncols() == 0 and mat.nrows() == 0
if equal:
return 0
return -1
def homology(self, dim=None, **kwds):
"""
The homology of the chain complex in the given dimension.
INPUT:
- ``dim`` - an element of the grading group for the chain
complex (optional, default None): the degree in which to
compute homology. If this is None, return the homology in
every dimension in which the chain complex is possibly
nonzero.
- ``base_ring`` - a commutative ring (optional, default is the
base ring for the chain complex). Must be either the
integers `\\ZZ` or a field.
- ``generators`` - boolean (optional, default False). If
True, return generators for the homology groups along with
the groups. NOTE: this is only implemented if the CHomP
package is available.
- ``verbose`` - boolean (optional, default False). If True,
print some messages as the homology is computed.
- ``algorithm`` - string (optional, default 'auto'). The
options are 'auto', 'dhsw', 'pari' or 'no_chomp'. See
below for descriptions.
OUTPUT: if dim is specified, the homology in dimension dim.
Otherwise, the homology in every dimension as a dictionary
indexed by dimension.
ALGORITHM:
If ``algorithm`` is set to 'auto' (the default), then use
CHomP if available. (CHomP is available at the web page
http://chomp.rutgers.edu/. It is also an experimental package
for Sage.)
CHomP computes homology, not cohomology, and only works over
the integers or finite prime fields. Therefore if any of
these conditions fails, or if CHomP is not present, or if
``algorithm`` is set to 'no_chomp', go to plan B: if ``self``
has a ``_homology`` method -- each simplicial complex has
this, for example -- then call that. Such a method implements
specialized algorithms for the particular type of cell
complex.
Otherwise, move on to plan C: compute the chain complex of
``self`` and compute its homology groups. To do this: over a
field, just compute ranks and nullities, thus obtaining
dimensions of the homology groups as vector spaces. Over the
integers, compute Smith normal form of the boundary matrices
defining the chain complex according to the value of
``algorithm``. If ``algorithm`` is 'auto' or 'no_chomp', then
for each relatively small matrix, use the standard Sage
method, which calls the Pari package. For any large matrix,
reduce it using the Dumas, Heckenbach, Saunders, and Welker
elimination algorithm: see
:func:`sage.homology.chain_complex.dhsw_snf` for details.
Finally, ``algorithm`` may also be 'pari' or 'dhsw', which
forces the named algorithm to be used regardless of the size
of the matrices and regardless of whether CHomP is available.
As of this writing, CHomP is by far the fastest option,
followed by the 'auto' or 'no_chomp' setting of using the
Dumas, Heckenbach, Saunders, and Welker elimination algorithm
for large matrices and Pari for small ones.
.. warning::
This only works if the base ring is the integers or a
field. Other values will return an error.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C.homology()
{0: Z x Z, 1: Z x C3}
sage: C.homology(dim=1, base_ring = GF(3))
Vector space of dimension 2 over Finite Field of size 3
sage: D = ChainComplex({0: identity_matrix(ZZ, 4), 4: identity_matrix(ZZ, 30)})
sage: D.homology()
{0: 0, 1: 0, 4: 0, 5: 0}
Generators (only available via CHomP): generators are given as
a list of cycles, each of which is an element in the
appropriate free module, and hence is represented as a vector::
sage: C.homology(1, generators=True) # optional - CHomP
(Z x C3, [(1, 0), (0, 1)])
"""
from sage.interfaces.chomp import have_chomp, homchain
algorithm = kwds.get('algorithm', 'auto')
verbose = kwds.get('verbose', False)
base_ring = kwds.get('base_ring', None)
if base_ring is None or base_ring == self._base_ring:
change_ring = False
base_ring = self._base_ring
else:
change_ring = True
if (not base_ring.is_field()) and (base_ring != ZZ):
raise NotImplementedError, "Can't compute homology if the base ring is not the integers or a field."
if (algorithm == 'auto' and (base_ring == ZZ or
(base_ring.is_prime_field() and base_ring != QQ))):
H = None
if have_chomp('homchain'):
H = homchain(self, **kwds)
if H:
if dim is not None:
field = base_ring.is_prime_field()
answer = {}
if isinstance(dim, (list, tuple)):
for d in dim:
if d in H:
answer[d] = H[d]
else:
if field:
answer[d] = VectorSpace(base_ring, 0)
else:
answer[d] = HomologyGroup(0)
else:
if dim in H:
answer = H[dim]
else:
if field:
answer = VectorSpace(base_ring, 0)
else:
answer = HomologyGroup(0)
else:
answer = H
return answer
else:
if verbose:
print "ran CHomP, but no output."
if algorithm not in ['dhsw', 'pari', 'auto', 'no_chomp']:
raise NotImplementedError, "algorithm not recognized"
if dim is None:
answer = {}
for n in self._diff.keys():
if verbose:
print "Computing homology of the chain complex in dimension %s..." % n
if base_ring == self._base_ring:
answer[n] = self.homology(n, verbose=verbose,
algorithm=algorithm)
else:
answer[n] = self.homology(n, base_ring=base_ring,
verbose=verbose,
algorithm=algorithm)
return answer
degree = self._degree
if not dim in self._grading_group:
raise ValueError, "The dimension does not appear to be an element of the grading group."
if dim in self._diff:
d_out_cols = self._diff[dim].ncols()
d_out_rows = self._diff[dim].nrows()
if min(d_out_cols, d_out_rows) == 0:
d_out_rank = 0
if base_ring == ZZ:
temp_ring = QQ
else:
temp_ring = base_ring
if (dim, temp_ring) in self._ranks:
d_out_rank = self._ranks[(dim, temp_ring)]
else:
if min(d_out_cols, d_out_rows) == 0:
d_out_rank = 0
elif change_ring or base_ring == ZZ:
d_out_rank = self._diff[dim].change_ring(temp_ring).rank()
else:
d_out_rank = self._diff[dim].rank()
self._ranks[(dim, temp_ring)] = d_out_rank
if dim-degree in self._diff:
if change_ring:
d_in = self._diff[dim-degree].change_ring(base_ring)
else:
d_in = self._diff[dim-degree]
if base_ring.is_field():
if d_out_cols == 0:
null = 0
elif d_out_rows == 0:
null = d_out_cols
else:
null = d_out_cols - d_out_rank
if d_in.nrows() == 0 or d_in.ncols() == 0:
rk = 0
else:
if (dim-degree, temp_ring) in self._ranks:
rk = self._ranks[(dim-degree, temp_ring)]
else:
rk = d_in.rank()
self._ranks[(dim-degree, temp_ring)] = rk
answer = VectorSpace(base_ring, null - rk)
elif base_ring == ZZ:
nullity = d_out_cols - d_out_rank
if d_in.ncols() == 0:
all_divs = [0] * nullity
else:
if algorithm == 'auto' or algorithm == 'no_chomp':
if ((d_in.ncols() > 300 and d_in.nrows() > 300)
or (min(d_in.ncols(), d_in.nrows()) > 100 and
d_in.ncols() + d_in.nrows() > 600)):
algorithm = 'dhsw'
else:
algorithm = 'pari'
if algorithm == 'dhsw':
all_divs = dhsw_snf(d_in, verbose=verbose)
else:
if d_in.is_sparse():
all_divs = d_in.dense_matrix().elementary_divisors(algorithm)
else:
all_divs = d_in.elementary_divisors(algorithm)
all_divs = all_divs[:nullity]
divisors = filter(lambda x: x != 1, all_divs)
answer = HomologyGroup(len(divisors), divisors)
else:
pass
else:
if base_ring.is_field():
answer = VectorSpace(base_ring, d_out_cols - d_out_rank)
elif base_ring == ZZ:
nullity = d_out_cols - d_out_rank
answer = HomologyGroup(nullity)
else:
pass
else:
if base_ring.is_field():
answer = VectorSpace(base_ring, 0)
elif base_ring == ZZ:
answer = HomologyGroup(0)
else:
answer = FreeModule(self._base_ring, rank=0)
if verbose:
print " Homology is %s" % answer
return answer
def betti(self, dim=None, **kwds):
"""
The Betti number of the homology of the chain complex in this
dimension.
That is, write the homology in this dimension as a direct sum
of a free module and a torsion module; the Betti number is the
rank of the free summand.
INPUT:
- ``dim`` - an element of the grading group for the chain
complex or None (optional, default None). If None, then
return every Betti number, as a dictionary indexed by
degree. If an element of the grading group, then return
the Betti number in that dimension.
- ``base_ring`` - a commutative ring (optional, default is the
base ring for the chain complex). Compute homology with
these coefficients. Must be either the integers or a
field.
OUTPUT: the Betti number in dimension ``dim`` - the rank of
the free part of the homology module in this dimension.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C.betti(0)
2
sage: [C.betti(n) for n in range(5)]
[2, 1, 0, 0, 0]
sage: C.betti()
{0: 2, 1: 1}
"""
base_ring = kwds.get('base_ring', None)
if base_ring is None:
base_ring = self._base_ring
if base_ring == ZZ:
base_ring = QQ
if base_ring.is_field():
kwds['base_ring'] = base_ring
H = self.homology(dim=dim, **kwds)
if isinstance(H, dict):
return dict([i, H[i].dimension()] for i in H)
else:
return H.dimension()
else:
raise NotImplementedError, "Not implemented: unable to compute Betti numbers if the base ring is not ZZ or a field."
def torsion_list(self, max_prime, min_prime=2):
"""
Look for torsion in this chain complex by computing its mod `p`
homology for a range of primes `p`.
INPUT:
- ``max_prime`` - prime number: search for torsion mod `p` for
all `p` strictly less than this number.
- ``min_prime`` - prime (optional, default 2): search for
torsion mod `p` for primes at least as big as this.
Return a list of pairs (`p`, ``dims``) where `p` is a prime at
which there is torsion and ``dims`` is a list of dimensions in
which this torsion occurs.
The base ring for the chain complex must be the integers; if
not, an error is raised.
Algorithm: let `C` denote the chain complex. Let `P` equal
``max_prime``. Compute the mod `P` homology of `C`, and use
this as the base-line computation: the assumption is that this
is isomorphic to the integral homology tensored with
`\\GF{P}`. Then compute the mod `p` homology for a range of
primes `p`, and record whenever the answer differs from the
base-line answer.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C.homology()
{0: Z x Z, 1: Z x C3}
sage: C.torsion_list(11)
[(3, [1])]
sage: C = ChainComplex([matrix(ZZ, 1, 1, [2]), matrix(ZZ, 1, 1), matrix(1, 1, [3])])
sage: C.homology(1)
C2
sage: C.homology(3)
C3
sage: C.torsion_list(5)
[(2, [1]), (3, [3])]
"""
if self._base_ring != ZZ:
raise ValueError, "This only works if the base ring of the chain complex is the integers"
else:
answer = []
torsion_free = self.betti(base_ring=GF(max_prime))
for p in prime_range(min_prime, max_prime):
mod_p_betti = self.betti(base_ring=GF(p))
if mod_p_betti != torsion_free:
diff_dict = {}
temp_diff = {}
D = self._degree
for i in torsion_free:
temp_diff[i] = mod_p_betti.get(i, 0) - torsion_free[i]
for i in temp_diff:
if temp_diff[i] > 0:
if i+D in diff_dict:
lower = diff_dict[i+D]
else:
lower = 0
current = temp_diff[i]
if current > lower:
diff_dict[i] = current - lower
if i-D in diff_dict:
diff_dict[i-D] -= current - lower
differences = []
for i in diff_dict:
if diff_dict[i] != 0:
differences.append(i)
answer.append((p,differences))
return answer
def category(self):
"""
Return the category to which this chain complex belongs: the
category of all chain complexes over the base ring.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])}, base_ring=GF(7))
sage: C.category()
Category of chain complexes over Finite Field of size 7
"""
import sage.categories.all
return sage.categories.all.ChainComplexes(self.base_ring())
def _Hom_(self, other, category=None):
"""
Return the set of chain maps between chain complexes ``self``
and ``other``.
EXAMPLES::
sage: S = simplicial_complexes.Sphere(2)
sage: T = simplicial_complexes.Torus()
sage: C = S.chain_complex(augmented=True,cochain=True)
sage: D = T.chain_complex(augmented=True,cochain=True)
sage: Hom(C,D) # indirect doctest
Set of Morphisms from Chain complex with at most 4 nonzero terms over Integer Ring to Chain complex with at most 4 nonzero terms over Integer Ring in Category of chain complexes over Integer Ring
"""
from sage.homology.chain_complex_homspace import ChainComplexHomspace
return ChainComplexHomspace(self, other)
def _flip_(self):
"""
Flip chain complex upside down (degree `n` gets changed to
degree `-n`), thus turning a chain complex into a cochain
complex without changing the homology (except for flipping it,
too).
EXAMPLES::
sage: C = ChainComplex({2: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C._degree
1
sage: C.differential(2)
[3 0 0]
[0 0 0]
sage: C._flip_()._degree
-1
sage: C._flip_().differential(-2)
[3 0 0]
[0 0 0]
"""
data = {}
deg = self._degree
for d in self.differential():
data[-d] = self.differential()[d]
return ChainComplex(data, degree=-deg)
def _chomp_repr_(self):
r"""
String representation of self suitable for use by the CHomP
program.
Since CHomP can only handle chain complexes, not cochain
complexes, and since it likes its complexes to start in degree
0, flip the complex over if necessary, and shift it to start
in degree 0. Note also that CHomP only works over the
integers or a finite prime field.
EXAMPLES::
sage: C = ChainComplex({-2: matrix(ZZ, 1, 3, [3, 0, 0])}, degree=-1)
sage: C._chomp_repr_()
'chain complex\n\nmax dimension = 1\n\ndimension 0\n boundary a1 = 0\n\ndimension 1\n boundary a1 = + 3 * a1 \n boundary a2 = 0\n boundary a3 = 0\n\n'
sage: C = ChainComplex({-2: matrix(ZZ, 1, 3, [3, 0, 0])}, degree=1)
sage: C._chomp_repr_()
'chain complex\n\nmax dimension = 1\n\ndimension 0\n boundary a1 = 0\n\ndimension 1\n boundary a1 = + 3 * a1 \n boundary a2 = 0\n boundary a3 = 0\n\n'
"""
if (self._grading_group != ZZ or
(self._degree != 1 and self._degree != -1)):
raise ValueError, "CHomP only works on Z-graded chain complexes with differential of degree 1 or -1."
base_ring = self._base_ring
if (base_ring == QQ) or (base_ring != ZZ and not (base_ring.is_prime_field())):
raise ValueError, "CHomP doesn't compute over the rationals, only over Z or F_p."
if self._degree == -1:
diffs = self.differential()
else:
diffs = self._flip_().differential()
maxdim = max(diffs)
mindim = min(diffs)
s = "chain complex\n\nmax dimension = %s\n\n" % (maxdim - mindim,)
for i in range(0, maxdim - mindim + 1):
s += "dimension %s\n" % i
mat = diffs.get(i + mindim, matrix(base_ring, 0, 0))
for idx in range(mat.ncols()):
s += " boundary a%s = " % (idx + 1)
col = mat.column(idx)
if col.nonzero_positions():
for j in col.nonzero_positions():
entry = col[j]
if entry > 0:
sgn = "+"
else:
sgn = "-"
entry = -entry
s += "%s %s * a%s " % (sgn, entry, j+1)
else:
s += "0"
s += "\n"
s += "\n"
return s
def _repr_(self):
"""
Print representation.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C._repr_()
'Chain complex with at most 2 nonzero terms over Integer Ring'
"""
diffs = filter(lambda mat: mat.nrows() + mat.ncols() > 0,
self._diff.values())
string1 = "Chain complex with at most"
string2 = " %s nonzero terms over %s" % (len(diffs),
self._base_ring)
return string1 + string2
def _latex_(self):
"""
LaTeX print representation.
EXAMPLES::
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])})
sage: C._latex_()
'\\Bold{Z}^{3} \\xrightarrow{d_{0}} \\Bold{Z}^{2}'
"""
string = ""
dict = self._diff
deg = self._degree
ring = self._base_ring
if self._grading_group != ZZ:
guess = dict.keys()[0]
if guess-deg in dict:
string += "\\dots \\xrightarrow{d_{%s}} " % latex(guess-deg)
string += _latex_module(ring, mat.ncols())
string += " \\xrightarrow{d_{%s}} \\dots" % latex(guess)
else:
backwards = (deg < 0)
sorted_list = sorted(dict.keys(), reverse=backwards)
if len(dict) <= 6:
for n in sorted_list[:-1]:
mat = dict[n]
string += _latex_module(ring, mat.ncols())
string += " \\xrightarrow{d_{%s}} " % latex(n)
mat = dict[sorted_list[-1]]
string += _latex_module(ring, mat.ncols())
else:
for n in sorted_list[:2]:
mat = dict[n]
string += _latex_module(ring, mat.ncols())
string += " \\xrightarrow{d_{%s}} " % latex(n)
string += "\\dots "
n = sorted_list[-2]
string += "\\xrightarrow{d_{%s}} " % latex(n)
mat = dict[sorted_list[-1]]
string += _latex_module(ring, mat.ncols())
return string
class HomologyGroup_class(AdditiveAbelianGroup_fixed_gens):
"""
Abelian group on `n` generators. This class inherits from
``AdditiveAbelianGroup``; see that for more documentation. The main
difference between the classes is in the print representation.
EXAMPLES::
sage: from sage.homology.chain_complex import HomologyGroup
sage: G = AbelianGroup(5,[5,5,7,8,9]); G
Multiplicative Abelian Group isomorphic to C5 x C5 x C7 x C8 x C9
sage: H = HomologyGroup(5,[5,5,7,8,9]); H
C5 x C5 x C7 x C8 x C9
sage: G == loads(dumps(G))
True
sage: AbelianGroup(4)
Multiplicative Abelian Group isomorphic to Z x Z x Z x Z
sage: HomologyGroup(4)
Z x Z x Z x Z
sage: HomologyGroup(100)
Z^100
"""
def __init__(self, n, invfac):
"""
See ``HomologyGroup`` for full documentation.
EXAMPLES::
sage: from sage.homology.chain_complex import HomologyGroup
sage: H = HomologyGroup(5,[5,5,7,8,9]); H
C5 x C5 x C7 x C8 x C9
"""
n = len(invfac)
A = ZZ**n
B = A.span([A.gen(i) * invfac[i] for i in xrange(n)])
AdditiveAbelianGroup_fixed_gens.__init__(self, A, B, A.gens())
self._original_invts = invfac
def _repr_(self):
"""
Print representation
EXAMPLES::
sage: from sage.homology.chain_complex import HomologyGroup
sage: H = HomologyGroup(7,[4,4,4,4,4,7,7])
sage: H._repr_()
'C4^5 x C7 x C7'
sage: HomologyGroup(6)
Z^6
"""
eldv = self._original_invts
if len(eldv) == 0:
return "0"
rank = len(filter(lambda x: x == 0, eldv))
torsion = sorted(filter(lambda x: x, eldv))
if rank > 4:
g = ["Z^%s" % rank]
else:
g = ["Z"] * rank
if len(torsion) != 0:
printed = []
for t in torsion:
numfac = torsion.count(t)
too_many = (numfac > 4)
if too_many:
if t not in printed:
g.append("C%s^%s" % (t, numfac))
printed.append(t)
else:
g.append("C%s" % t)
times = " x "
return times.join(g)
def _latex_(self):
"""
LaTeX representation
EXAMPLES::
sage: from sage.homology.chain_complex import HomologyGroup
sage: H = HomologyGroup(7,[4,4,4,4,4,7,7])
sage: H._latex_()
'C_{4}^{5} \\times C_{7} \\times C_{7}'
sage: latex(HomologyGroup(6))
\ZZ^{6}
"""
eldv = self._original_invts
if len(eldv) == 0:
return "0"
rank = len(filter(lambda x: x == 0, eldv))
torsion = sorted(filter(lambda x: x, eldv))
if rank > 4:
g = ["\\ZZ^{%s}" % rank]
else:
g = ["\\ZZ"] * rank
if len(torsion) != 0:
printed = []
for t in torsion:
numfac = torsion.count(t)
too_many = (numfac > 4)
if too_many:
if t not in printed:
g.append("C_{%s}^{%s}" % (t, numfac))
printed.append(t)
else:
g.append("C_{%s}" % t)
times = " \\times "
return times.join(g)
def HomologyGroup(n, invfac=None):
"""
Abelian group on `n` generators. This class inherits from
``AdditiveAbelianGroup``; see that for more documentation. The main
difference between the classes is in the print representation.
EXAMPLES::
sage: from sage.homology.chain_complex import HomologyGroup
sage: G = AbelianGroup(5,[5,5,7,8,9]); G
Multiplicative Abelian Group isomorphic to C5 x C5 x C7 x C8 x C9
sage: H = HomologyGroup(5,[5,5,7,8,9]); H
C5 x C5 x C7 x C8 x C9
sage: AbelianGroup(4)
Multiplicative Abelian Group isomorphic to Z x Z x Z x Z
sage: HomologyGroup(4)
Z x Z x Z x Z
sage: HomologyGroup(100)
Z^100
"""
if invfac is None:
if isinstance(n, (list, tuple)):
invfac = n
n = len(n)
else:
invfac = []
if len(invfac) < n:
invfac = [0] * (n - len(invfac)) + invfac
elif len(invfac) > n:
raise ValueError, "invfac (=%s) must have length n (=%s)"%(invfac, n)
M = HomologyGroup_class(n, invfac)
return M
