Theorem 1

Let $A$ be a ring and $P$ an $A$*-module*. The following properties are equivalent:

(a) $P$ is a finitely generated projective module. (b) $P$ is a finitely presented module and, for every maximal ideal $m$ of $A$, $P_m$ is a free $A_m$*-module*. (c) $P$ is a finitely generated module, for all $p \in \mathrm{Spec}(A)$, the $A_p$*-module* $P_p$ is free and, if we denote its rank by $r_p$, the function $p \mapsto r_p$ is locally constant in the topological space $\mathrm{Spec}(A)$ (that is, every point of $\mathrm{Spec}(A)$ admits a neighbourhood in which this function is constant).

(d) There exists a finite family $(f_i)_{i \in I}$ of elements of $A$, generating the ideal $A$, such that, for all $i \in I$, the $A,$-module $P_{f_i}$ is free with finite rank.

(e) For every maximal ideal $m$ of $A$, there exists $f \in A - m$ such that $P_f$ is a free $A,$-module of finite rank.

We show the theorem by proving the following scheme of implications

(a) ⇔ (d)

(b) ⇌ (e)

(c)

(a) ⇒ (b): We know that a finitely generated projective module is finitely presented (Chapter I, § 2, no. 8, Lemma 8 (iii)); if $P$ is a projective $A$-module, $P_m = P \otimes_A A_m$ is a projective $A_m$-module (\emph{Algebra}, Chapter II, § 5, no. 1, Corollary to Proposition 4); finally, as $A_m$ is a local ring, every finitely presented projective $A_m$-module is free (§ 3, no. 2, Corollary to Proposition 5).

(b) ⇒ (e): This follows from the Corollary to Proposition 2 of no. 1.

(c) ⇒ (e): Let $m$ be a maximal ideal of $A$; write $r_m = n$ and let $(x_i)_{1 \leq i \leq n}$ be a basis of $P$. We can assume that the $x_i$ are canonical images of elements $p_i \in P$ $(1 \leq i \leq n)$ to within multiplication by an invertible element of $A,$. Let $(e_i)_{1 \leq i \leq n}$ be the canonical basis of $A^n$ and $u : A^n \to P$ the homomorphism such that $u(e_i) = p_i$ for $1 \leq i \leq n$. As $P$ is finitely generated, it follows from Proposition 2 of no. 1 that there exists $f \in A - m$ such that $u_f$ is surjective. We conclude that $u_{f,g}$ is also surjective for all $g \in A - m$ and by hypothesis there exists $g \in A - m$ such that $r_p = n$ for $p \in X,$. Then, replacing $f$ by $f g$, we may assume that $r_p = n$ for all $p \in X,$. Then $u_p : A_p^n \to P_p$ is a surjective homomorphism and $P_p$ and $A_p$ are both free $A,$-modules of the same rank; hence (§ 3, no. 2, Corollary to Proposition 6) $u_p$ is bijective for all $p \in X_f$. Let $p'$ be a prime ideal of $A,$ and let $p$ be its inverse image in $A$ under the canonical mapping; if $(A_f^{n})_{p'}$ and $(P_f)_{p'}$ are identified with $A_p^n$ and $P_p$ under the canonical isomorphisms, $(u_f)_{p'}$ is identified with $u_p$ and is consequently bijective. We conclude that $u_f$ is bijective (§ 3, no. 3, Theorem 1), which establishes (e).

(e) ⇒ (d): Let $E$ be the set off $\in A$ such that $P_f$ is a finitely generated free $A_f$-module. The hypothesis implies that $E$ is contained in no maximal ideal of $A,$ hence $E$ generates the ideal $A$ and there therefore exist a finite family $(f_i)_{1 \leq i \leq n}$ of elements of $E$ and $a_i \in A$ $(1 \leq i \leq n)$ such that $1 = \sum_{i=1}^n a_i f_i$; whence (d).

(d) ⇒ (c): It follows from no. 1, Corollary to Proposition 3 that $P$ is finitely generated. On the other hand, for every prime ideal $p$ of $A$, there exists an index $i$ such that $p \in X_{f_i}$; if $p' = p_{f_i}$, then $P_p = (P_{f_i})_{p'}$ (§ 2, no. 5, Proposition

10) and hence by hypothesis $P_p$ is free and of the same rank as $P_{f_i}$, which proves (c).

(d) ⇒ (a): Consider the ring $B = \prod_{i \in I} A_i$ and the B-module

For every index $i$, there exists a free $A_i$-module $L$, such that $P_i$ is a direct factor of $L$, and it may be assumed that all the $L$, have the same rank; then $L = \prod_{i \in I} L_i$ is a free B-module of which M is a direct factor, in other words M is a finitely generated projective B-module. As B is a faithfully flat A-module (no. 1, Proposition 3), we conclude that P is a finitely generated projective A-module (Chapter I, §3, no. 6, Proposition 12).