Timeline for Confusing definitions in Liu's Algebraic geometry and arithmetic curves?
Current License: CC BY-SA 2.5
17 events
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| Jul 19, 2010 at 7:57 | vote | accept | user717 | ||
| Jul 19, 2010 at 7:57 | comment | added | user717 | @Qing: I was pretty sure about that :) I hope you don't take it personal that I keep mixing up your first and last name :) | |
| Jul 18, 2010 at 22:26 | comment | added | Qing Liu | Dear Georges, merci pour les compliments ! Dear Arminus, there is no problem critizing my book, and you are wellcome to do it. As I am keep writing an errata, it will help me to improve the book. | |
| Jul 18, 2010 at 16:18 | comment | added | Georges Elencwajg | Dear Arminius, it was quite clear to me that you weren't criticizing Qing: I just wanted to state that MY answer wasn't meant as criticism either. I quite agree that it is unfair to downvote you: I don't advertise my upvotes normally (there are 251), but in this case let me tell you exceptionally that I did upvote you two hours ago. | |
| Jul 18, 2010 at 15:36 | comment | added | user717 | choice. So the style of definition in Quing's book (which gives the definition in a restricted situation which is probably easier) indeed makes sense as long as one mentions the extra conditions all along. | |
| Jul 18, 2010 at 15:34 | comment | added | user717 | @Georges: Moreover, thanks to your link, I think I understand, that my proposal for an alternative definition isn't the right way to do it. Take for example the definition of a smooth morphism in Quing's book: Let $Y$ be locally noetherian and let $f:X \rightarrow Y$ be a morphism of finite type. [...]. We say that $f$ is smooth if [...]. Now, I would have changed that to "a morphism $f:X \rightarrow Y$ of schemes is called smooth if it is of finite type, if $Y$ is locally noetherian and if [...]". But due to the more general definition given in the stacks project, this wouldn't be a good | |
| Jul 18, 2010 at 15:18 | comment | added | user717 | @Georges: The intention of the second point in my edit above was to make clear that I'm not criticizing Liu Quing, his book, or algebraic geometry in general, and that my questions just emerged from getting confused by some definitions and knowing about experts here who can explain how I have to deal with those. Getting down voted for asking questions about a mathematical textbook (let it be the best written book on earth; I am telling people to take a look at Quing's book instead of Hartshorne's for quite some time now by the way) is something that produces confusion, too. | |
| Jul 18, 2010 at 14:52 | comment | added | Georges Elencwajg | Arminius's questions are of course completely legitimate and interesting, but I'd like to emphasize that Qing's book is very well-written and contains an amazing wealth of material. I bought it in 2003 but I am still very far, alas, from having mastered it... | |
| Jul 18, 2010 at 14:21 | history | edited | user717 | CC BY-SA 2.5 |
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| Jul 18, 2010 at 14:17 | comment | added | Qing Liu | Continued: I think in the book, the word birational is never used out of this situation. For 3), yes I should just restrict to noetherian and not locally noetherian schemes. It is already so hard to desingularize noetherian schemes :). Yes by definition (EGA IV), desingularization morphism are proper, so in particular are of finite type. | |
| Jul 18, 2010 at 14:14 | comment | added | Qing Liu | As for 1), the explanation is as given by Victor. As far as I know, regularity is defined only for locally noetherian schemes, so regular scheme are supposed to be locally noetherian, as least in my book. So yes, maybe it is better so say a scheme is regular if it is locally noetherian and etc. For 2), the reason is different. I only consider birational finite type morphism over a reduced noetherian scheme. Contrarily to 1), as pointed out by Georges, it can be defined in a much more general setting. Here I can not say a morphism if birational if it is finite type etc. | |
| Jul 18, 2010 at 13:51 | answer | added | Georges Elencwajg | timeline score: 6 | |
| Jul 18, 2010 at 13:32 | comment | added | Andrea Ferretti | Yet it is strange that in 3) a weaker property (local Noetherianity) than an implicit one (Noetherianity) is explicitly mentioned. | |
| Jul 18, 2010 at 13:18 | comment | added | Victor Protsak | As to "why" only the author can provide a definite answer. Although he seems to be participating in MO, it would appear to be less argumentative if you asked him directly. I personally have always thought that "Let X have property P. We say that X is Q if ..." means that in talking about X that are Q, we only consider X with P (seems a bit tautological). It's a great devise for not dragging along monstrous sequences of standard assumptions (kind of like saying "in this book, all rings are commutative and have an identity"). | |
| Jul 18, 2010 at 12:55 | history | edited | user717 | CC BY-SA 2.5 |
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| Jul 18, 2010 at 12:38 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Jul 18, 2010 at 12:13 | history | asked | user717 | CC BY-SA 2.5 |