Cochain complex

Definition

Let $R$ be a ring and let $\{C^n\}_{n\in\mathbb Z}$ be R-modules . A cochain complex $(C^\bullet, d)$ is a collection of $R$-linear maps

such that

Its cohomology modules are

see cohomology module .

Cross-links

• 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 .

Examples

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 $n\neq 0$.

2. 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

   $$\delta^n(\varphi) := \varphi\circ d_{n+1} \in \operatorname{Hom}_R(C_{n+1},M).$$

   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 .

3. “Multiplication by $x$” as a cochain complex.
   For $x\in R$, the 2-term cochain complex

   $$0 \to R \xrightarrow{\,\cdot x\,} R \to 0$$

   (degrees $0\to 1$) has

   $$H^0 \cong \operatorname{Ann}_R(x),\qquad H^1 \cong R/xR.$$