Chain complexes¶
This module implements bounded 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. The elements of the grading group are generally called degrees, so \(C_n\) is the module in degree \(n\) and so on.
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_of_differential
parameter used in defining the chain
complex.
AUTHORS:
John H. Palmieri (2009-04): initial implementation
- sage.homology.chain_complex.ChainComplex(data=None, base_ring=None, grading_group=None, degree_of_differential=1, degree=1, check=True)¶
Define a chain complex.
INPUT:
data
– the data defining the chain complex; see below for more details.
The following keyword arguments are supported:
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 additive free abelian group (optional, defaultZZ
), the group over which the chain complex is indexed.degree_of_differential
– element of grading_group (optional, default1
). The degree of the differential.degree
– alias fordegree_of_differential
.check
– boolean (optional, defaultTrue
). IfTrue
, 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.
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 \(C_{n+d}\).)a list/tuple/iterable 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 \(\ZZ\) anddegree
is 1.a list/tuple/iterable of the form \([r_0, d_0, r_1, d_1, r_2, d_2, \ldots]\), 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 \(\ZZ\) anddegree
is 1.a list/tuple/iterable of the form \([d_0, d_1, d_2, \ldots]\) where each \(d_i\) is a matrix, as above. This only makes sense if
grading_group
is \(\ZZ\) anddegree
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 ascheck
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() Trivial chain complex 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: m = matrix(ZZ, 2, 2, [0, 1, 0, 0]) sage: D = ChainComplex([m, m], 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 sage: D.differential(0)==m, m.is_immutable(), D.differential(0).is_immutable() (True, False, 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: diff = IZ.differential() # the differentials in the chain complex sage: diff[-1], diff[0], diff[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: no common canonical parent for objects with parents: 'Finite Field in a of size 5^3' and 'Rational Field'
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 convert 0 to a rational
- class sage.homology.chain_complex.ChainComplex_class(grading_group, degree_of_differential, base_ring, differentials)¶
Bases:
sage.structure.parent.Parent
See
ChainComplex()
for full documentation.The differentials are required to be in the following canonical form:
All differentials that are not \(0 \times 0\) must be specified (even if they have zero rows or zero columns), and
Differentials that are \(0 \times 0\) must not be specified.
Immutable matrices over the
base_ring
This and more is ensured by the assertions in the constructor. The
ChainComplex()
factory function must ensure that only valid input is passed.EXAMPLES:
sage: C = ChainComplex(); C Trivial chain complex 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
- Element¶
alias of
Chain_class
- betti(deg=None, base_ring=None)¶
The Betti number of the chain complex.
That is, write the homology in this degree as a direct sum of a free module and a torsion module; the Betti number is the rank of the free summand.
INPUT:
deg
– an element of the grading group for the chain complex or None (defaultNone
); ifNone
, then return every Betti number, as a dictionary indexed by degree, or if an element of the grading group, then return the Betti number in that degreebase_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 degree
deg
– the rank of the free part of the homology module in this degree.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} sage: D = ChainComplex({0:matrix(GF(5), [[3, 1],[1, 2]])}) sage: D.betti() {0: 1, 1: 1}
- cartesian_product(*factors, **kwds)¶
Return the direct sum (Cartesian product) of
self
withD
.Let \(C\) and \(D\) be two chain complexes with differentials \(\partial_C\) and \(\partial_D\), respectively, of the same degree (so they must also have the same grading group). The direct sum \(S = C \oplus D\) is a chain complex given by \(S_i = C_i \oplus D_i\) with differential \(\partial = \partial_C \oplus \partial_D\).
INPUT:
subdivide
– (default:False
) whether to subdivide the the differential matrices
EXAMPLES:
sage: R.<x,y> = QQ[] sage: C = ChainComplex([matrix([[-y],[x]]), matrix([[x, y]])]) sage: D = ChainComplex([matrix([[x-y]]), matrix([[0], [0]])]) sage: ascii_art(C.cartesian_product(D)) [x y 0] [ -y 0] [0 0 0] [ x 0] [0 0 0] [ 0 x - y] 0 <-- C_2 <-------- C_1 <-------------- C_0 <-- 0 sage: D = ChainComplex({1:matrix([[x-y]]), 4:matrix([[x], [y]])}) sage: ascii_art(D) [x] [y] [x - y] 0 <-- C_5 <---- C_4 <-- 0 <-- C_2 <-------- C_1 <-- 0 sage: ascii_art(cartesian_product([C, D])) [-y] [x] [ x y 0] [ x] [y] [ 0 0 x - y] [ 0] 0 <-- C_5 <---- C_4 <-- 0 <-- C_2 <-------------------- C_1 <----- C_0 <-- 0
The degrees of the differentials must agree:
sage: C = ChainComplex({1:matrix([[x]])}, degree_of_differential=-1) sage: D = ChainComplex({1:matrix([[x]])}, degree_of_differential=1) sage: C.cartesian_product(D) Traceback (most recent call last): ... ValueError: the degrees of the differentials must match
- degree_of_differential()¶
Return the degree of the differentials of the complex
OUTPUT:
An element of the grading group.
EXAMPLES:
sage: D = ChainComplex({0: matrix(ZZ, 2, 2, [1,0,0,2])}) sage: D.degree_of_differential() 1
- differential(dim=None)¶
The differentials which make up the chain complex.
INPUT:
dim
– element of the grading group (optional, defaultNone
); if this isNone
, return a dictionary of all of the differentials, or 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] sage: D.differential(-1) [] sage: C = ChainComplex({0: identity_matrix(ZZ, 40)}) sage: diff = C.differential() sage: diff[-1] 40 x 0 dense matrix over Integer Ring (use the '.str()' method to see the entries) sage: diff[0] 40 x 40 dense matrix over Integer Ring (use the '.str()' method to see the entries) sage: diff[1] []
- dual()¶
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_of_differential() 1 sage: C.differential(2) [3 0 0] [0 0 0] sage: C.dual().degree_of_differential() -1 sage: C.dual().differential(3) [3 0] [0 0] [0 0]
- free_module(degree=None)¶
Return the free module at fixed
degree
, or their sum.INPUT:
degree
– an element of the grading group orNone
(default).
OUTPUT:
The free module \(C_n\) at the given degree \(n\). If the degree is not specified, the sum \(\bigoplus C_n\) is returned.
EXAMPLES:
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0]), 1: matrix(ZZ, [[0, 1]])}) sage: C.free_module() Ambient free module of rank 6 over the principal ideal domain Integer Ring sage: C.free_module(0) Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: C.free_module(1) Ambient free module of rank 2 over the principal ideal domain Integer Ring sage: C.free_module(2) Ambient free module of rank 1 over the principal ideal domain Integer Ring
- free_module_rank(degree)¶
Return the rank of the free module at the given
degree
.INPUT:
degree
– an element of the grading group
OUTPUT:
Integer. The rank of the free module \(C_n\) at the given degree \(n\).
EXAMPLES:
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0]), 1: matrix(ZZ, [[0, 1]])}) sage: [C.free_module_rank(i) for i in range(-2, 5)] [0, 0, 3, 2, 1, 0, 0]
- grading_group()¶
Return the grading group.
OUTPUT:
The discrete abelian group that indexes the individual modules of the complex. Usually \(\ZZ\).
EXAMPLES:
sage: G = AdditiveAbelianGroup([0, 3]) sage: C = ChainComplex(grading_group=G, degree=G(vector([1,2]))) sage: C.grading_group() Additive abelian group isomorphic to Z + Z/3 sage: C.degree_of_differential() (1, 2)
- homology(deg=None, base_ring=None, generators=False, verbose=False, algorithm='pari')¶
The homology of the chain complex.
INPUT:
deg
– an element of the grading group for the chain complex (default:None
); the degree in which to compute homology – if this isNone
, return the homology in every degree 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 fieldgenerators
– boolean (optional, defaultFalse
); ifTrue
, return generators for the homology groups along with the groups. See trac ticket #6100verbose
- boolean (optional, defaultFalse
); ifTrue
, print some messages as the homology is computedalgorithm
- string (optional, default'pari'
); the options are:'auto'
'chomp'
'dhsw'
'pari'
'no_chomp'
see below for descriptions
OUTPUT:
If the degree is specified, the homology in degree
deg
. Otherwise, the homology in every dimension as a dictionary indexed by dimension.ALGORITHM:
If
algorithm
is set to'auto'
, then use CHomP if available. CHomP is available at the web page http://chomp.rutgers.edu/. It is also an optional package for Sage. Ifalgorithm
ischomp
, always use chomp.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: ifself
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 ofalgorithm
. Ifalgorithm
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 [DHSW2003]: seedhsw_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,
'pari'
is the fastest standard option. The optional CHomP package may be better still.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(deg=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: 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. Each summand of the homology is listed separately, with a corresponding generator:
sage: C.homology(1, generators=True, algorithm='no_chomp') [(C3, Chain(1:(1, 0))), (Z, Chain(1:(0, 1)))]
Note that the answer will be formatted differently if the optional package CHomP is installed.
sage: C.homology(1, generators=True) # optional - CHomP (Z x C3, [(0, 1), (1, 0)])
Tests for trac ticket #6100, the Klein bottle with generators:
sage: d0 = matrix(ZZ, 0,1) sage: d1 = matrix(ZZ, 1,3, [[0,0,0]]) sage: d2 = matrix(ZZ, 3,2, [[1,1], [1,-1], [-1,1]]) sage: C_k = ChainComplex({0:d0, 1:d1, 2:d2}, degree=-1) sage: C_k.homology(generators=true, algorithm='no_chomp') {0: [(Z, Chain(0:(1)))], 1: [(C2, Chain(1:(0, 1, -1))), (Z, Chain(1:(0, 1, 0)))], 2: []}
From a torus using a field:
sage: T = simplicial_complexes.Torus() sage: C_t = T.chain_complex() sage: C_t.homology(base_ring=QQ, generators=True) {0: [(Vector space of dimension 1 over Rational Field, Chain(0:(0, 0, 0, 0, 0, 0, 1)))], 1: [(Vector space of dimension 1 over Rational Field, Chain(1:(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1))), (Vector space of dimension 1 over Rational Field, Chain(1:(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0)))], 2: [(Vector space of dimension 1 over Rational Field, Chain(2:(1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1)))]}
- nonzero_degrees()¶
Return the degrees in which the module is non-trivial.
See also
ordered_degrees()
.OUTPUT:
The tuple containing all degrees \(n\) (grading group elements) such that the module \(C_n\) of the chain is non-trivial.
EXAMPLES:
sage: one = matrix(ZZ, [[1]]) sage: D = ChainComplex({0: one, 2: one, 6:one}) sage: ascii_art(D) [1] [1] [0] [1] 0 <-- C_7 <---- C_6 <-- 0 ... 0 <-- C_3 <---- C_2 <---- C_1 <---- C_0 <-- 0 sage: D.nonzero_degrees() (0, 1, 2, 3, 6, 7)
- ordered_degrees(start=None, exclude_first=False)¶
Sort the degrees in the order determined by the differential
INPUT:
start
– (default:None
) a degree (element of the grading group) orNone
exclude_first
– boolean (optional; default:False
); whether to exclude the lowest degree – this is a handy way to just get the degrees of the non-zero modules, as the domain of the first differential is zero.
OUTPUT:
If
start
has been specified, the longest tuple of degreescontaining
start
(unlessstart
would be the first andexclude_first=True
),in ascending order relative to
degree_of_differential()
, andsuch that none of the corresponding differentials are \(0\times 0\).
If
start
has not been specified, a tuple of such tuples of degrees. One for each sequence of non-zero differentials. They are returned in sort order.EXAMPLES:
sage: one = matrix(ZZ, [[1]]) sage: D = ChainComplex({0: one, 2: one, 6:one}) sage: ascii_art(D) [1] [1] [0] [1] 0 <-- C_7 <---- C_6 <-- 0 ... 0 <-- C_3 <---- C_2 <---- C_1 <---- C_0 <-- 0 sage: D.ordered_degrees() ((-1, 0, 1, 2, 3), (5, 6, 7)) sage: D.ordered_degrees(exclude_first=True) ((0, 1, 2, 3), (6, 7)) sage: D.ordered_degrees(6) (5, 6, 7) sage: D.ordered_degrees(5, exclude_first=True) (6, 7)
- random_element()¶
Return a random element.
EXAMPLES:
sage: D = ChainComplex({0: matrix(ZZ, 2, 2, [1,0,0,2])}) sage: D.random_element() # random output Chain with 1 nonzero terms over Integer Ring
- rank(degree, ring=None)¶
Return the rank of a differential
INPUT:
degree
– an element \(\delta\) of the grading group. Which differential \(d_{\delta}\) we want to know the rank ofring
– (optional) a commutative ring \(S\); if specified, the rank is computed after changing to this ring
OUTPUT:
The rank of the differential \(d_{\delta} \otimes_R S\), where \(R\) is the base ring of the chain complex.
EXAMPLES:
sage: C = ChainComplex({0:matrix(ZZ, [[2]])}) sage: C.differential(0) [2] sage: C.rank(0) 1 sage: C.rank(0, ring=GF(2)) 0
- shift(n=1)¶
Shift this chain complex \(n\) times.
INPUT:
n
– an integer (optional, default 1)
The shift operation is also sometimes called translation or suspension.
To shift a chain complex by \(n\), shift its entries up by \(n\) (if it is a chain complex) or down by \(n\) (if it is a cochain complex); that is, shifting by 1 always shifts in the opposite direction of the differential. In symbols, if \(C\) is a chain complex and \(C[n]\) is its \(n\)-th shift, then \(C[n]_j = C_{j-n}\). The differential in the shift \(C[n]\) is obtained by multiplying each differential in \(C\) by \((-1)^n\).
Caveat: different sources use different conventions for shifting: what we call \(C[n]\) might be called \(C[-n]\) in some places. See for example. https://ncatlab.org/nlab/show/suspension+of+a+chain+complex (which uses \(C[n]\) as we do but acknowledges \(C[-n]\)) or 1.2.8 in [Wei1994] (which uses \(C[-n]\)).
EXAMPLES:
sage: S1 = simplicial_complexes.Sphere(1).chain_complex() sage: S1.shift(1).differential(2) == -S1.differential(1) True sage: S1.shift(2).differential(3) == S1.differential(1) True sage: S1.shift(3).homology(4) Z
For cochain complexes, shifting goes in the other direction. Topologically, this makes sense if we grade the cochain complex for a space negatively:
sage: T = simplicial_complexes.Torus() sage: co_T = T.chain_complex()._flip_() sage: co_T.homology() {-2: Z, -1: Z x Z, 0: Z} sage: co_T.degree_of_differential() 1 sage: co_T.shift(2).homology() {-4: Z, -3: Z x Z, -2: Z}
You can achieve the same result by tensoring (on the left, to get the signs right) with a rank one free module in degree
-n * deg
, ifdeg
is the degree of the differential:sage: C = ChainComplex({-2: matrix(ZZ, 0, 1)}) sage: C.tensor(co_T).homology() {-4: Z, -3: Z x Z, -2: Z}
- tensor(*factors, **kwds)¶
Return the tensor product of
self
withD
.Let \(C\) and \(D\) be two chain complexes with differentials \(\partial_C\) and \(\partial_D\), respectively, of the same degree (so they must also have the same grading group). The tensor product \(S = C \otimes D\) is a chain complex given by
\[S_i = \bigoplus_{a+b=i} C_a \otimes D_b\]with differential
\[\partial(x \otimes y) = \partial_C x \otimes y + (-1)^{|a| \cdot |\partial_D|} x \otimes \partial_D y\]for \(x \in C_a\) and \(y \in D_b\), where \(|a|\) is the degree of \(a\) and \(|\partial_D|\) is the degree of \(\partial_D\).
Warning
If the degree of the differential is even, then this may not result in a valid chain complex.
INPUT:
subdivide
– (default:False
) whether to subdivide the the differential matrices
Todo
Make subdivision work correctly on multiple factors.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: C1 = ChainComplex({1:matrix([[x]])}, degree_of_differential=-1) sage: C2 = ChainComplex({1:matrix([[y]])}, degree_of_differential=-1) sage: C3 = ChainComplex({1:matrix([[z]])}, degree_of_differential=-1) sage: ascii_art(C1.tensor(C2)) [ x] [y x] [-y] 0 <-- C_0 <------ C_1 <----- C_2 <-- 0 sage: ascii_art(C1.tensor(C2).tensor(C3)) [ y x 0] [ x] [-z 0 x] [-y] [z y x] [ 0 -z -y] [ z] 0 <-- C_0 <-------- C_1 <----------- C_2 <----- C_3 <-- 0
sage: C = ChainComplex({2:matrix([[-y],[x]]), 1:matrix([[x, y]])}, ....: degree_of_differential=-1); ascii_art(C) [-y] [x y] [ x] 0 <-- C_0 <------ C_1 <----- C_2 <-- 0 sage: T = C.tensor(C) sage: T.differential(1) [x y x y] sage: T.differential(2) [-y x 0 y 0 0] [ x 0 x 0 y 0] [ 0 -x -y 0 0 -y] [ 0 0 0 -x -y x] sage: T.differential(3) [ x y 0 0] [ y 0 -y 0] [-x 0 0 -y] [ 0 y x 0] [ 0 -x 0 x] [ 0 0 x y] sage: T.differential(4) [-y] [ x] [-y] [ x]
The degrees of the differentials must agree:
sage: C1p = ChainComplex({1:matrix([[x]])}, degree_of_differential=1) sage: C1.tensor(C1p) Traceback (most recent call last): ... ValueError: the degrees of the differentials must match
- torsion_list(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 numbermin_prime
– prime (optional, default 2); search for torsion mod \(p\) for primes at least as big as this
Return a list of pairs \((p, d)\) where \(p\) is a prime at which there is torsion and \(d\) 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])]
- class sage.homology.chain_complex.Chain_class(parent, vectors, check=True)¶
Bases:
sage.structure.element.ModuleElement
A Chain in a Chain Complex
A chain is collection of module elements for each module \(C_n\) of the chain complex \((C_n, d_n)\). There is no restriction on how the differentials \(d_n\) act on the elements of the chain.
Note
You must use the chain complex to construct chains.
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
- is_boundary()¶
Return whether the chain is a boundary.
OUTPUT:
Boolean. Whether the elements of the chain are in the image of the differentials.
EXAMPLES:
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])}) sage: c = C({0:vector([0, 1, 2]), 1:vector([3, 4])}) sage: c.is_boundary() False sage: z3 = C({1:(1, 0)}) sage: z3.is_cycle() True sage: (2*z3).is_boundary() False sage: (3*z3).is_boundary() True
- is_cycle()¶
Return whether the chain is a cycle.
OUTPUT:
Boolean. Whether the elements of the chain are in the kernel of the differentials.
EXAMPLES:
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])}) sage: c = C({0:vector([0, 1, 2]), 1:vector([3, 4])}) sage: c.is_cycle() True
- vector(degree)¶
Return the free module element in
degree
.EXAMPLES:
sage: C = ChainComplex({0: matrix(ZZ, 2, 3, [3, 0, 0, 0, 0, 0])}) sage: c = C({0:vector([1, 2, 3]), 1:vector([4, 5])}) sage: c.vector(0) (1, 2, 3) sage: c.vector(1) (4, 5) sage: c.vector(2) ()