Timeline for Is a quasi-coherent sheaf of ideals with free stalks of rank 1 a Cartier divisor?
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17 events
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| Dec 30, 2014 at 4:06 | comment | added | Question Mark | @user74230: That's a good idea! I'll add the neron-models tag to all my questions about the book. (I hope these phantom edits won't result in too much spam on top of the site.) | |
| Dec 30, 2014 at 4:02 | history | edited | Question Mark |
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| Dec 30, 2014 at 2:12 | comment | added | user74230 | @QuestionMark: To make the MO postings more useful for future users seeking such stuff I recommend that you put the label "neron-models" for all of these questions (not just "algebraic-geometry"); I haven't looked back at the earlier ones but this one is only labeled with ag, but I think such a heron-models tag does exist. | |
| Dec 29, 2014 at 17:47 | comment | added | Question Mark | @user74230: It would be good to have this included in SP where you suggest, but first someone needs to come up with a counterexample (which is what I was hoping for with this question; I've tried inventing one, but didn't succeed). Thanks for the comment about (i) of Lemma 6. | |
| Dec 29, 2014 at 16:33 | comment | added | Question Mark | @JasonStarr: I have posted all the more serious glitches that I have noticed while reading. The ones I have not posted are either typos or modifications to some proofs needed to fix minor glitches (I haven't posted the latter because they don't affect the validity of any Theorems/Propositions). Overall, as you know, this book is very solid in all aspects, especially, in handling highly technical details, and there are very few errors. | |
| Dec 29, 2014 at 16:25 | comment | added | Question Mark | @user74230: I post these questions because I think it is useful for potential further readers or aficionados of the book to be aware of the small errors present and modifications that need to be made. Most of these are harmless, but some are genuinely confusing or somewhat serious gaps (not this one in particular); most would also be overlooked by most readers. Since these "questions" are so specific that many readers may not have a real person to consult with (most people can't be bothered about small details anyway), why not have them available on MO (I can't think of a better medium)? | |
| Dec 29, 2014 at 16:22 | comment | added | user74230 | @QM: I agree fully about the value of recording "known pathologies"; you may wish to consider suggesting that deJong include this one about the invertible ideal sheaves in his Museum of Horrors (aka the chapter Examples) in the Stacks Project. In Lemma 6, with $\mathscr{I}$ just finite type, (i) is OK as written since $S$-flatness of $D$ at $x$ and f.p. over $S$ for $D$ and lfp for $X$ over $S$ force $\mathscr{I}$ to be f.p. over $O_X$ by "descent to the noetherian case" (by openness of flatness for lfp maps, such as $D\rightarrow S$). | |
| Dec 29, 2014 at 16:14 | comment | added | user74230 | @JasonStarr: Certainly no harm. I'm just puzzled why these are being posted so often individually on MO rather than compiled as a single document and discussed once afterwards (to have more perspective on what is important). There are small harmless errors like the above in many standard books (e.g., Hartshorne's AG textbook) and it isn't clear that posting questions about them one-by-one on MO is appropriate. | |
| Dec 29, 2014 at 16:13 | comment | added | Question Mark | @user74230: I agree, but a conterexample would nevertheless be an interesting thing to have (to complement other known pathologies in algebraic geometry). By the way, for latter purposes Lemma 6 on p. 213 needs to be upgraded (same proof): $\mathscr{I}$ should only be assumed of loc. finite type and, in (i), invertible at a neighborhood of $x$ (with the parenthetical pertaining only to the finite presentation case). This "upgrade" is needed on p. 215 to get $\mathrm{Div}_{X/S}$ an open subfunctor of $\mathrm{Hilb}_{X/S}$ ($D$ there being f.p. only means that $\mathscr{I}_D$ is of f.t.). | |
| Dec 29, 2014 at 15:59 | vote | accept | Question Mark | ||
| Dec 29, 2014 at 14:33 | comment | added | Jason Starr | @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? | |
| Dec 29, 2014 at 14:26 | comment | added | user74230 | @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. | |
| Dec 29, 2014 at 14:10 | comment | added | Jason Starr | @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. | |
| Dec 29, 2014 at 12:46 | answer | added | jmc | timeline score: 3 | |
| Dec 29, 2014 at 6:58 | comment | added | 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 $\mathscr{I}$ is finitely presented (loc. cit., Proposition 2). | |
| Dec 29, 2014 at 6:53 | comment | added | user74230 | 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. :) | |
| Dec 28, 2014 at 19:51 | history | asked | Question Mark | CC BY-SA 3.0 |