Exact complex - Imported Knowlpedia Content

A chain complex $(C_\bullet,d)$ is exact if any (hence all) of the following equivalent conditions hold:

$H_n(C_\bullet)=0$ for all $n$ , where homology module is used.
For every $n$ , $\operatorname{im}(d_{n+1}) = \ker(d_n).$
The sequence $\cdots \xrightarrow{d_{n+2}} C_{n+1} \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1}\xrightarrow{d_{n-1}} \cdots$ is exact at each $C_n$ in the sense of kernels/images; compare exactness via kernels and images.

In an abelian category, the same definition makes sense using categorical kernels and images.

Remarks

Exactness for short sequences: short exact sequence.
A strong way to prove exactness is via a contracting homotopy: chain homotopy.
Exact complexes appear as resolutions: projective resolution and injective resolution.

Examples

A short exact sequence as an exact complex. Any short exact sequence $0\to A\to B\to C\to 0$ of $R$ -modules is an exact complex with $A,B,C$ placed in degrees $1,0,-1$ (or $2,1,0$ , depending on convention). See exact sequence of modules.

Identity map complex. The 2-term complex $0 \to R \xrightarrow{\,\mathrm{id}\,} R \to 0$ is exact: $\ker(\mathrm{id})=0$ and $\operatorname{coker}(\mathrm{id})=0$ . In fact, it is contractible (see chain homotopy example), so its homology vanishes.

A standard exact sequence over $\mathbb Z$ . For $n\ge 1$ , the sequence $0 \longrightarrow \mathbb Z \xrightarrow{\,\cdot n\,} \mathbb Z \longrightarrow \mathbb Z/n\mathbb Z \longrightarrow 0$ is short exact, hence an exact complex. This 2-term free resolution of $\mathbb Z/n\mathbb Z$ is the starting point for concrete computations of Tor and Ext.