Let $X$ be a scheme and let $\mathscr{I} \subset \mathscr{O}\_X$ be a quasi-coherent sheaf of ideals. Suppose that for each $x \in X$, the stalk $\mathscr{I}\_x$ is generated by an element $f\_x \in \mathscr{O}\_x$ that comes from a nonzero divisor $f \in \Gamma(U, \mathscr{O}\_X)$ for some affine open neighborhood $U$ of $x$ (so $f\_x$ is a nonzero divisor, too). Then each $\mathscr{I}\_x$ is a free $\mathscr{O}\_x$-module of rank $1$. Is $\mathscr{I}$ necessarily locally free of rank $1$?


The answer is surely 'yes' if $X$ is Noetherian (e.g., by <http://stacks.math.columbia.edu/tag/0AG8>), but a positive answer is claimed in general in the last paragraph on p. 212 of the book "Neron models". Is there a counterexample to this claim?



Is a quasi-coherent sheaf of ideals with free stalks of rank 1 a Cartier divisor?
CW answer to get this of the unanswered list. Answered in the comments:



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> **user74230:** The authors simply forgot to assume that I is of finite presentation as an OX-module. What ultimately matters is Lemma 6 on p. 213, so don't take that claim on p. 212 too seriously as written. When you write a 300-page book on technically difficult mathematics, good luck making no harmless minor glitches like that. :)
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> **abx:** Bourbaki, Commutative Algebra, chap. II, § 5, exercise 7. As indicated in the previous comment, the answer is also 'yes' with no noetherian hypothesis if I is finitely presented (loc. cit., Proposition 2).
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