exact complex

Definition

A complex is exact if its homology .

What does that mean?

A complex is exact if and only if for all . So we have the following simple cases

• is exact if and only if is injective.
• is exact if and only if is surjective.
• is exact if and only if is injective with cokernel and is surjective with kernel . An exact complex of this form is called a short exact sequence.

Example

Any morphism of R-modules fits into an exact complex