cochain complex

A cochain complex is the same idea as a chain complex but with the arrows pointing the other way: instead of degrees dropping by one along each map, they rise by one. Think of it as climbing a staircase rather than descending it. This bookkeeping choice matters because cohomology — the version of homology built from cochain complexes — is where products and other extra structure naturally live.

Concretely, a cochain complex over a ring R is a sequence ... -> C^{n-1} -> C^n -> C^{n+1} -> ... with maps d^n : C^n -> C^{n+1} (the coboundary maps) satisfying d^{n+1} ∘ d^n = 0. Elements of ker(d^n) are n-cocycles; elements of im(d^{n-1}) are n-coboundaries. The condition d^2 = 0 again forces every coboundary to be a cocycle, and exactness at C^n means im(d^{n-1}) = ker(d^n).

Whether one writes a chain or a cochain complex is purely a matter of indexing convention: relabeling C^n as C_{-n} turns one into the other. The distinction is meaningful in practice because the natural complexes that arise differ in direction — the singular chains of a space form a chain complex, while applying Hom or taking differential forms produces cochain complexes. Cohomology of such a complex is the cohomology in the topological or algebraic theory at hand.