Examples of "cochain"
The cochain groups can be made into a cochain complex formula_19 by defining the coboundary operator
For example, a 1-cochain "f" is a 1-coboundary if there exists a 0-cochain "h" such that formula_30
Alexander had by 1930 defined a first notion of a cochain, by thinking of an "i"-cochain on a space "X" as a function on small neighborhoods of the diagonal in "X".
Thus a (q−1)-cochain "f" is a cocycle if for all "q"-simplices σ the cocycle condition formula_27 holds. In particular, a 1-cochain "f" is a 1-cocycle if
The construction starts with a product of cochains: if "c" is a "p"-cochain and
defined by contracting a singular chain formula_2 with a singular cochain formula_3 by the formula :
satisfying formula_40. This is a cochain complex known as the "de Rham complex".
One may check that formula_8, so this defines a cochain complex whose cohomology can be computed.
There is an analogous definition using injective resolutions and cochain complexes.
One important property of the exterior derivative is that . This means that the exterior derivative defines a cochain complex:
Some 2-cocycles can be obtained from 1-cochains. A 1-cochain on is simply a linear map,
The Lie algebra cohomology of the Lie algebra formula_10 over the field formula_11, with values in the left formula_10-module formula_13 can be computed using the "Chevalley-Eilenberg complex" formula_14. The formula_15-cochains in this complex are the alternating formula_11-multilinear functions formula_17 of formula_15 variables with values in formula_13. The coboundary of an formula_15-cochain is the formula_21-cochain formula_22 given by
A variant on the concept of chain complex is that of "cochain complex". A cochain complex formula_3 is a sequence of abelian groups or modules ..., formula_4, formula_5, formula_6, formula_7, formula_8, ... connected by homomorphisms formula_9 such that the composition of any two consecutive maps is the zero map: formula_10 for all "n":
The index formula_12 in either formula_13 or formula_14 is referred to as the degree (or dimension). The only difference in the definitions of chain and cochain complexes is that, in chain complexes, the boundary operators decrease dimension, whereas in cochain complexes they increase dimension.
The elements of the individual groups of a chain complex are called chains (or cochains in the case of a cochain complex.) The image of "d" is the group of boundaries, or in a cochain complex, coboundaries. The kernel of "d" (i.e., the subgroup sent to 0 by "d") is the group of cycles, or in the case of a cochain complex, cocycles. From the basic relation, the (co)boundaries lie inside the (co)cycles. This phenomenon is studied in a systematic way using (co)homology groups.
An example of graded vector space is associated to a chain complex, or cochain complex "C" of vector spaces; the latter takes the form
On the other hand, the differential forms, with exterior derivative, , as the connecting map, form a cochain complex, which defines the de Rham cohomology groups .
and morphisms the chain maps. (It is equivalent to consider "cochain complexes" of objects of "A", where the numbering is written as
A "q"-cochain is called a "q"-cocycle if it is in the kernel of δ, hence formula_26 is the set of all "q"-cocycles.
A "q"-cochain is called a "q"-coboundary if it is in the image of "δ" and formula_29 is the set of all "q"-coboundaries.