Remark

If $f, g \in \mathrm{Hom}_A(M, N)$, one does not have $L(f + g) = L(f) + L(g)$. However these two morphisms are homotopic according to X, p. 49, prop. 3.

Let $M$ be a right a-module; denote by $A^\circ$ the opposite ring of $A$, by $M^\circ$ the underlying $A^\circ$-module of $M$, by $L(M^\circ)$ its canonical free resolution. This is denoted by $L(M)$, and the underlying A-complex $L(M^\circ)^\circ$ of $L(M^\circ)$ is called the canonical free resolution of $M$. Thus one has