Proposition 10

Let $M$ be an $A$-module. The following conditions are equivalent:

(i) $M$ is projective. (ii) $\operatorname{Ext}_A^i(M, N) = 0$ for every $A$-module $N$ and for every integer $i > 0$. (iii) $\operatorname{Ext}_A^1(M, N) = 0$ for every $A$-module $N$. (iv) There exists an exact sequence

where $P$ is projective, and where $\operatorname{Ext}_A^1(M, K) = 0$.

(i) $\Rightarrow$ (ii) : this is the corollary of prop. 5 of X, p. 88. (ii) $\Rightarrow$ (iii) : this is trivial. (iii) $\Rightarrow$ (iv) : this is clear since $M$ is quotient of a free module $P$. (iv) $\Rightarrow$ (i) : since $\operatorname{Ext}_A^1(M, K) = 0$, the canonical mapping

is surjective (X, p. 90, corollary); hence there exists an $A$-linear section of $\nu$ and $M$ is isomorphic to a direct factor of $P$, and therefore is projective.