Definition 1

Let $M$ be an $A$-module. A left resolution of $M$ is a pair $(P,p)$ where $P$ is a complex zero on the right and $p:P\to M$ is a homomorphism. A right resolution of $M$ is a pair $(e,E)$ where $E$ is a complex zero on the left and $e:M\to E$ a homomorphism.

The length of the resolution $(P,p)$ (resp. $(e,E)$) is called the length of the complex $P$ (resp. $E$). If $(P,p)$ and $(P',p')$ (resp. $(e,E)$ and $(e',E')$) are two left (resp. right) resolutions of $M$, a morphism of complexes $f:P\to P'$ such that $p'\circ f=p$ (resp. $g:E\to E'$ such that $g\circ e=e'$) is called a morphism of resolutions.