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!