Chain complexes, chain maps, and homotopies

A cochain differential raises degree by one, and two successive differentials compose to zero. Consequently every boundary is a cycle, so the quotient defining cohomology exists. This requirement must be checked before calculating any quotient: arbitrary consecutive matrices need not form a complex. A chain differential instead lowers degree. Setting $C^n=C_{-n}$ converts a chain complex into a cochain complex without altering its individual arrows; homology in degree $i$ becomes cohomology in degree $-i$.

In a finite-dimensional complex over a field, cohomology measures the gap between the kernel of one matrix and the image of the preceding matrix. Over integers, ranks alone miss torsion in that quotient. For example, multiplication by a nonzero integer has zero kernel but usually a nontrivial cokernel. Recording the ring is therefore part of the calculation, rather than an optional label: multiplication by six behaves differently over integers, rational numbers, and a field of characteristic three.

A cochain map consists of maps in every degree satisfying $d_D^nf^n=f^{n+1}d_C^n$. This identity sends cycles to cycles and boundaries to boundaries, giving a well-defined map on cohomology. Two degreewise linear maps that happen to give the same matrix rank are not thereby maps of complexes. Compatibility is an equation involving both adjacent degrees, and an alleged induced map cannot be used before that equation is established.

A cochain homotopy from one map to another is a family lowering degree by one. On a cycle, their difference becomes a boundary, which proves that homotopic maps induce identical cohomology maps. The converse is false in general: knowing an induced map discards chain-level information. Homotopy is stronger data than agreement on cohomology, and its formula explains precisely where the discarded difference goes instead of merely asserting that the difference is invisible.

A homotopy equivalence has a cochain-map inverse up to homotopy in both directions. It is necessarily a quasi-isomorphism, meaning an isomorphism on every cohomology module. A contraction is a homotopy between the identity and zero. It forces all cohomology to vanish, since every cycle is explicitly the differential of its contracted image. Acyclicity only says that each cycle has some preimage; it does not supply compatible module homomorphisms choosing those preimages.

The distinction matters over rings where short exact sequences need not split. Over a field, choices of complements allow an acyclic complex of vector spaces to be contracted. Over integers, an exact complex containing the nonsplit quotient onto the integers modulo two can be acyclic without being contractible. Later, derived localization will deliberately erase such acyclic complexes. First we retain them, because their failure to contract explains why homotopy and cohomology describe different levels of equivalence.