Exact complex

Definition

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

1. $H_n(C_\bullet)=0$ for all $n$, where homology module is used.
2. For every $n$, $\operatorname{im}(d_{n+1}) = \ker(d_n).$
3. 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.

Cross-links

• 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

1. 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 .

2. 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.

3. 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 .