Proposition 9

Let $M$ be an $A$-module and $(P,p)$ a projective resolution of $M$. Let $r$ denote the radical of $A$. Suppose either that $P_n$ is a finitely generated $A$-module for every $n$, or that $r$ is nilpotent. Then a necessary and sufficient condition for $(P,p)$ to be minimal is that the complex $(A/r)\otimes_A P$ have zero differential, in other words that

Suppose that $(P,p)$ is minimal. By VIII, loc. cit., the homomorphism

is an isomorphism. It then follows from the exact sequence

that the homomorphism $1 \otimes j_n : (A/r) \otimes_A \operatorname{Im} d_{n+1} \to (A/r) \otimes_A P_n$ is zero; since $d_{n+1} = j_n \circ \delta_{n+1}$, we deduce that $1_{A/r} \otimes d_{n+1} = 0$ for $n \geqslant 0$.

Conversely, suppose that for every $n \geqslant 1$, $1 \otimes d_n$ is zero, in other words that $\operatorname{Im} d_n = \operatorname{Ker} d_{n-1}$ is contained in $rP_{n-1}$. Since $\delta_{n-1}$ is surjective, it follows from VIII, loc. cit. that $\delta_{n-1}$ is a projective cover for $n \geqslant 1$, hence that $(P, p)$ is minimal.