Exact sequence

Remarks

A sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is exact exactly when $f$ is injective, $g$ is surjective, and $\im f = \ker g$; then $C \cong B / f(A)$. Exactness of $0 \to A \xrightarrow{f} B$ means that $f$ is injective, and exactness of $B \xrightarrow{g} C \to 0$ means that $g$ is surjective. Exactness at $M_i$ implies $f_i \circ f_{i-1} = 0$, but not conversely.

Example: $0 \to \Z \xrightarrow{2} \Z \xrightarrow{\pi} \Z/2\Z \to 0$, with $\pi$ the reduction, is a short exact sequence of $\Z$-modules. Multiplication by $2$ is injective, $\pi$ is onto, and $\im(2) = 2\Z = \ker \pi$.

Non-example: the sequence $\Z \xrightarrow{2} \Z \xrightarrow{\rho} \Z/4\Z$, with $\rho$ the reduction modulo $4$, is not exact at the middle term. The image of the first map is $2\Z$, but $\ker \rho = 4\Z \ne 2\Z$.