exact complex

Definition

A complex $(M_{\bullet },f_{\bullet })$ is exact if its homology $H_{\bullet }(M_{\bullet },f_{\bullet })=0$.

What does that mean?

A complex is exact if and only if $\text{im }{f}_{i+1}=\ker f_i$ for all $i$. So we have the following simple cases

• $0\to U\stackrel{i}{\to }M$ is exact if and only if $i$ is injective.
• $M\stackrel{\pi }{\to }N\to 0$ is exact if and only if $\pi$ is surjective.
• $0\to U\stackrel{i}{\to }M\stackrel{\pi }{\to }N\to 0$ is exact if and only if $i$ is injective with cokernel $\pi$ and $\pi$ is surjective with kernel $i$. An exact complex of this form is called a short exact sequence.

Example

Any morphism of R-modules $f:M\to N$ fits into an exact complex

$$0⟶\ker f⟶M\stackrel{f}{\to }N⟶\text{coker }f⟶0$$