Theorem 2

Let $P$ be an $A$-module and $n$ an integer $\geqslant 0$. The following properties are equivalent:

(a) $P$ is projective of rank $n$.

(b) $P$ is finitely generated and, for every maximal ideal $m$ of $A$, the $A$-*module* $P$, is free of rank $n$.

(c) $P$ is finitely generated and, for every prime ideal $p$ of $A$, the $A$-*module* $P_p$ is free of rank $n$.

(d) For every maximal ideal $m$ of $A$, there exists $f \in A - m$ such that the $A$-*module* $P_f$ is free of rank $n$.

By Definition 2 and Theorem 1, (a) and (c) are equivalent; (b) implies (c), as, for every prime ideal $p$ of $A$, there exists a maximal ideal $m$ containing $p$ and, writing $p' = p_m$, $P_p$ is isomorphic to $(P_m)_{p'}$ (§ 2, no. 5, Proposition 11); if $P_m$ is free of rank $n$, so then is $P_{p'}$. Property (c) implies (d) by virtue of Theorem 1 and the fact that, iff $\in A - m$ and $m' = m_f$, $P_m$ is isomorphic to $(P_f)_{m'}$ and therefore the ranks of $P_f$ and $P$ are equal. Finally, this last argument and Theorem 1 show that (d) implies (b).