Cochain complex

Its cohomology modules are

see cohomology module.

• Chain vs. cochain conventions: compare chain complex.
• Cochain complexes from $\mathrm{Hom}$: Hom and left exactness of Hom.
• Cohomology as a derived functor: derived functor and Ext.

1. Cochain complex concentrated in degree 0. For an $R$-module $M$, the diagram
   $$\cdots \to 0 \to M \to 0 \to \cdots$$
   with $M$ in degree $0$ is a cochain complex. Then $H^0\cong M$ and $H^n=0$ for $n\neq 0$.

1. Hom of a chain complex is a cochain complex. If $(C_\bullet,d)$ is a chain complex and $M$ is an $R$-module, define
   $$\operatorname{Hom}_R(C_\bullet,M)^n := \operatorname{Hom}_R(C_n,M),$$
   and set
   $$\delta^n(\varphi) := \varphi\circ d_{n+1} \in \operatorname{Hom}_R(C_{n+1},M).$$
   These Hom groups form a cochain complex of abelian groups in general. If $R$ is commutative (or the Hom groups carry a specified compatible scalar action), they may be viewed as a cochain complex of $R$-modules. Then $\delta^{n+1}\circ \delta^n=0$ because $d\circ d=0$, so this is a cochain complex. This construction underlies the computation of Ext from a projective resolution.

1. “Multiplication by $x$” as a cochain complex. For a commutative ring $R$ and $x\in R$, the 2-term cochain complex
   $$0 \to R \xrightarrow{\,\cdot x\,} R \to 0$$
   (degrees $0\to 1$) has
   $$H^0 \cong \operatorname{Ann}_R(x),\qquad H^1 \cong R/xR.$$