Definition 5

Let F, G, H be three A-modules; let f be a homomorphism of F into G and g a homomorphism of G into H. The ordered pair (f, g) is called an exact sequence if

in other words, if the kernel of g is equal to the image of f.

The diagram

(12)

is also called an exact sequence.

We consider similarly a diagram consisting of four A-modules and three homomorphisms:

(13)

This diagram is called exact at F if the diagram $E \xrightarrow{f} F \xrightarrow{g} G$ is exact; it is called exact at G if $F \xrightarrow{g} G \xrightarrow{h} H$ is exact. If diagram (13) is exact at F and at G, it is simply called exact, or an exact sequence. Exact sequences with an arbitrary number of terms are defined similarly.