Cochain complex ( $(C^{∙}, d)$ )

A cochain complex is a sequence of abelian groups or modules and maps

d^{n} : C^{n} \to C^{n + 1}

satisfying

d^{n + 1} d^{n} = 0

. Its cohomology is

H^{n} (C) = ker d^{n} / im d^{n - 1}

.

Cochain map ( $d_{F} f = f d_{E}$ )

f : E \to F

consists of degreewise maps

f^{i} : E^{i} \to F^{i}

satisfying

d_{F}^{i} f^{i} = f^{i + 1} d_{E}^{i}

. This identity sends cocycles to cocycles and coboundaries to coboundaries, hence induces maps on cohomology. For complexes of sheaves the same identity holds as an identity of sheaf morphisms.

Shift of a cochain complex ( $Σ E = E [1]$ )

The shift

E [1]

has

E [1]^{i} = E^{i + 1}

and differential

- d_{E}^{i + 1}

. Thus

H^{i} (E [1]) = H^{i + 1} (E)

. In general the differential of

E [r]

is

(- 1)^{r} d_{E}

.

Cycle module ( $Z^{q} = ker d^{q}$ )

The submodule of cocycles in degree

q

. A projective boundary module

B^{q + 1}

makes

0 \to Z^{q} \to P^{q} \to B^{q + 1} \to 0

split when

P^{q}

is projective, so

Z^{q}

is then also projective.

Cocycle ( $z \in ker d^{q}$ )

An element killed by the next differential of a cochain complex. Its class modulo coboundaries is a cohomology class.

Boundary module ( $B^{q} = im d^{q - 1}$ )

The submodule of coboundaries in degree

q

. The cohomology group is the corresponding cycle module modulo the boundary module.

Coboundary ( $b \in im d^{q - 1}$ )

An element in the image of the preceding differential of a cochain complex. Every coboundary is a cocycle, since consecutive differentials compose to zero.

Filtered cochain complex ( $\dots \supseteq F^{p} C \supseteq F^{p + 1} C \supseteq \dots$ )

A cochain complex with a decreasing filtration by cochain subcomplexes. A filtration finite in each degree is bounded in the degreewise sense; uniform bounds

F^{a} C = C

and

F^{b} C = 0

are sufficient. The associated spectral sequence starts with the graded quotient complexes.

Cochain subcomplex ( $d (S^{n}) \subseteq S^{n + 1}$ )

A sequence of submodules preserved by the differential of a cochain complex. Its inclusion is a map of complexes.

Short exact sequence of cochain complexes ( $0 \to A^{} \to B^{} \to C^{*} \to 0$ )

A degreewise short exact sequence of cochain complexes induces a long exact sequence from a coefficient sequence in the coefficient case, and a long exact cohomology sequence generally. Its degree-one connecting homomorphism lifts a quotient cocycle, takes its coboundary, and identifies that result in the subcomplex.

Coboundary map ( $d$ )

A coboundary map is the degree-one differential

d^{n} : C^{n} \to C^{n + 1}

of a cochain complex.

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