Exact sequence (categorical)

Let $\mathcal A$ be an abelian category and consider a composable pair

A \xrightarrow{f} B \xrightarrow{g} C.

The sequence is exact at $B$ if the image of $f$ equals the kernel of $g$ as subobjects of $B$ :

\operatorname{im}(f)=\ker(g).

This condition implies $g\circ f=0$ .

In an abelian category one can define the image via kernels and cokernels:

\operatorname{im}(f) := \ker(\operatorname{coker}(f)),

A longer sequence

\cdots \to A_{i-1}\xrightarrow{d_{i-1}} A_i \xrightarrow{d_i} A_{i+1}\to \cdots

is exact if it is exact at every object $A_i$ , i.e. $\operatorname{im}(d_{i-1})=\ker(d_i)$ for all $i$ .

Short exact sequences

A sequence

0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0

is short exact if it is exact at $A$ , $B$ , and $C$ . In an abelian category this is equivalent to:

Here $0$ denotes a zero object.

Examples

In $\mathbf{Ab}$ : multiplication by $2$ . $0 \to 2\mathbb Z \xrightarrow{\ \iota\ } \mathbb Z \xrightarrow{\ \pi\ } \mathbb Z/2\mathbb Z \to 0,$ where $\iota$ is inclusion and $\pi$ is reduction mod $2$ . Exactness at $\mathbb Z$ says $\operatorname{im}(\iota)=2\mathbb Z=\ker(\pi)$ .

In $\mathbf{Ab}$ : $\mathbb Z\subset \mathbb Q$ . $0 \to \mathbb Z \to \mathbb Q \to \mathbb Q/\mathbb Z \to 0$ is short exact: the quotient $\mathbb Q/\mathbb Z$ measures how $\mathbb Q$ differs from $\mathbb Z$ .

In $R$ $-$ $\mathbf{Mod}$ : principal ideal quotient. For a commutative ring $R$ and an element $x\in R$ , $R \xrightarrow{\ \cdot x\ } R \to R/(x) \to 0$ is exact (and is short exact on the left if $\cdot x$ is injective). Here $(x)$ is the image of the multiplication map, so exactness at the middle $R$ says $\operatorname{im}(\cdot x)=\ker(R\twoheadrightarrow R/(x))$ .