Cochain complex ($(C^{\bullet },d)$)

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 $H^n(C)=\ker d^n/\mathrm{im}{d}^{n-1}$.

A cochain differential is a degree-one map in a cochain complex:The square-zero identity makes every coboundary a cocycle, so the quotient $H^n(C)=\ker d^n/\mathrm{im}{d}^{n-1}$ defining cohomology is meaningful. The maps are homomorphisms of the abelian groups, or module-linear maps, used in the complex.

For $C^0=\mathbb{Z}$, $C^1=\mathbb{Z}$, $d^0(n)=2n$, and all other groups zero, the identity is automatic. The degree-zero cohomology is zero, while the degree-one cohomology is $\mathbb{Z}/2\mathbb{Z}$: every degree-one element is closed, but only the even ones are exact.

The exterior derivative on differential forms is another cochain differential, with $d^2=0$. In contrast, a chain differential lowers degree. The word differential records this graded square-zero structure, not necessarily differentiation of scalar functions.

A cochain map $f:E\to F$ 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.

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 $(-1{)}^r{d}_E$.

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.

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

The submodule of coboundaries in degree $q$. The cohomology group is the corresponding cycle module modulo the boundary module.

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

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.

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

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.

A coboundary map is the degree-one differential $d^n:C^n\to C^{n+1}$ of a cochain complex.