# Is a quasi-coherent sheaf of ideals with free stalks of rank 1 a Cartier divisor?

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?

• The authors simply forgot to assume that $\mathscr{I}$ is of finite presentation as an $O_X$-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. :) – user74230 Dec 29 '14 at 6:53
• Bourbaki, Commutative Algebra, chap. II, § 5, exercise 7. As indicated in the previous comment, the answer is also 'yes' with no noetherian hypothesis if $\mathscr{I}$ is finitely presented (loc. cit., Proposition 2). – abx Dec 29 '14 at 6:58
• @user74230: Is there an erratum for that book? I love that book, and I direct my advisees to read it. If there are mistakes (apparently quite a few, according to Question Mark), I would like to know what they are. – Jason Starr Dec 29 '14 at 14:10
• @JasonStarr: I'm not aware of an erratum, but the rate of errors is no different than other books of comparable length/level and (as you know) its handling of technical issues is masterfully clean. I don't know why Q.M. posts so many questions about this book, since most of those "errors" do not affect anything of interest (e.g., the above incorrect "claim") and are easily bypassed. I noticed all of these when reading the book as a student and never found them to be a hindrance to understanding. – user74230 Dec 29 '14 at 14:26
• @user74230: Thanks for the reply. I have also never had a student so confused by any point in the book that we could not work out the solution in a few minutes. However, students are often confused by small issues, and it would make sense to collect those. I also find Question Mark curious, but if the final result is a thorough erratum, what is the harm? – Jason Starr Dec 29 '14 at 14:33