Exact sequence (categorical)

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

The sequence is exact at $B$ if:

1. $g\circ f = 0$, and
2. the image of $f$ equals the kernel of $g$ as subobjects of $B$:
   $$\operatorname{im}(f) \;=\; \ker(g).$$

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

using kernels and cokernels.

A longer sequence

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

A sequence

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

• $f$ is a monomorphism,
• $g$ is a epimorphism,
• $\operatorname{im}(f)=\ker(g)$.

(Here $0$ denotes a zero object; compare additive category.)

1. 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 $\operatorname{im}(\iota)=2\mathbb Z=\ker(\pi)$.

1. 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 $\mathbb Z$.

1. In $R$$-$$\mathbf{Mod}$: principal ideal quotient. For a 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 $\operatorname{im}(\cdot x)=\ker(R\twoheadrightarrow R/(x))$.