Exact sequence of modules

A sequence of module homomorphisms where each image equals the next kernel.

Exact sequence of modules

An exact sequence of modules is a sequence of modules and module homomorphisms

\cdots \to M_{i-1}\xrightarrow{d_{i-1}} M_i \xrightarrow{d_i} M_{i+1}\to \cdots

such that for every $i$ one has $\operatorname{im}(d_{i-1})=\ker(d_i)$ , where the kernel and image are taken in the sense of kernels and images . A convenient checklist formulation is given in exactness via kernels/images .

Exact sequences package algebraic information: injectivity, surjectivity, quotients, and splitting phenomena are all phrased as exactness conditions.

Examples:

For a submodule $N\le M$ , the sequence $0\to N\to M\to M/N\to 0$ is exact.
The sequence $0\to \mathbb Z \xrightarrow{\times n} \mathbb Z \to \mathbb Z/n\mathbb Z \to 0$ is exact for $n\ne 0$ .
(Edge case) The sequence $M\xrightarrow{\mathrm{id}} M\to 0$ is exact, but $0\to M\xrightarrow{0} M$ is not exact unless $M=0$ .