If the chain is complex, all terms except finite numbers are zero, and all non-zero terms are finite generating commutative groups (or finite dimensional vector spaces), then Euler characteristics can be defined.

(commutative group adopts order, and vector space adopts Hamel dimension). In fact, it can also be calculated at the homology level:

In addition, especially in algebraic topology, this provides two important invariants for calculating the objects that produce chain complexes.

Short exact sequence of each chain complex

Resulting in a long exact sequence of homologous groups.

All the mappings in this long exact sequence are derived from the mappings between chain complexes, except the mappings. The latter is called connected homomorphism and is given by snake Lemma.