In Bourbaki's Commutative Algebra we have the following theorem:


II.5.2 Let $A$ ba a ring and $P$ an $A$-module. TFAE:


(i) $P$ is a f.g. projective module\


(ii) $P$ is a finitely presented module and, for every maximal ideal $m$ of $A$, $P\_m$ is a free $A\_m$-module.\


(iii) $P$ is a f.g. module, for all $p \in$ Spec($A$), the $\_p$-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 Spec($A$).


There is more to this theorem but I have stated what I need for my question. Namely, unless I am missing something, for (iii) implies (i) don't we need the extra condition that $A$ is a reduced ring? I see their proof and cannot find the problem but Eisenbud suggests that there exists a counterexample in problem 20.13 in Commutative Algebra with a View Toward Algebraic Geometry. I must be missing something!



On Bourbaki's characterization of projectives...
Dear fishibones, an important point is that at the beginning of Chapter II, Bourbaki states that all the rings he will consider are commutative. This implies that if a free module has finite dimension, this dimension is unambiguously defined. This property is called the Invariant Basis Number property (IBN) and may fail for non-commutative rings (Counterexamples can be found in, say, Lam's Lectures on Modules and Rings, Springer GTM 189.) This unambiguously defined dimension of a finitely generated free module is what Bourbaki calls its *rank*.


Let me emphasize that Bourbaki only uses "rank" in the above sense i.e. for finitely generated free modules. If $M$ is such a module of rank $r$ over the ring $A$, then for any prime ideal $\frak {p}$$ \in A$ the modules
$M\_{\frak p}$ over $A\_{\frak {p}}$ and $M\_{\frak {p}} \otimes\_{A\_{\frak p}} \kappa (p)$ over $\kappa (p)$
also are free of rank $r$. With this definition and use of rank, Bourbaki's Théorème 1 that you mention is perfectly correct (bien sûr!)


There is no contradiction with Eisenbud's exercise: he does *not* assume that his module $M$ has all its localizations $M\_{\frak {p}}$ free over $R\_{\frak {p}}$, whereas Bourbaki does.
Eisenbud only assumes that the dimension of the $\kappa ({\frak p})$- vector space $M\_{\frak {p}} \otimes\_{R\_{\frak p}} \kappa (p)$ is locally constant, while Bourbaki does not even mention this dimension when $M\_{\frak {p}}$ is not $R\_{\frak {p}}$-free.


