Timeline for A finitely generated, locally free module over a domain which is not projective?
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
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| Jan 7, 2011 at 2:21 | comment | added | Clark Barwick | Georges, thanks! Given the remarkable quality of your work as well as your answers on this site, it's high praise indeed. Ying, sorry I missed this till now! You are right that in the 3rd Corollary I should have said "finitely generated ideal." | |
| Jan 5, 2011 at 20:34 | comment | added | Georges Elencwajg | Congratulations on your amazing erudition, Clark, I'm really impressed. | |
| Mar 18, 2010 at 23:04 | comment | added | Ying Zhang | Hi, I saw in some books that we are only talking about the relationship of being projective and locally free when the module M assumed to be finitely generated. Is it true that non-finitely generated locally free modules over commutative Noetherian domains are also projective? Also, in the third Corollary above, are we assuming the flat ideal I to be finitely generated? | |
| Feb 4, 2010 at 14:30 | comment | added | Clark Barwick | @Harry Gindi: It's the 10th chapter of Algèbre. I fixed the reference to make it clearer. @Pete: Thanks! I'm glad it's interesting! | |
| Feb 4, 2010 at 14:28 | history | edited | Clark Barwick | CC BY-SA 2.5 |
clarified reference, fixed typo
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| Feb 4, 2010 at 14:23 | comment | added | Pete L. Clark | @Everyone: This is a really, really good answer. It deserves more upvotes than the question. | |
| Feb 4, 2010 at 14:22 | comment | added | Harry Gindi | Is Algebre Homologique the 10th chapter of Algebre or Algebre Commutative? | |
| Feb 4, 2010 at 14:21 | comment | added | Clark Barwick | My apologies: I'd (mis)interpreted your question to mean that you were looking for a down-to-earth way of thinking about the difference between projectivity and finite generation plus flatness, so I was trying to describe a principled way of building new counterexamples. I edited the answer as you suggested. I don't think I've ever seen the result on integral domains before (and in the first draft of my answer, I wasn't even sure it was true!), but I don't see any reason for it to be excluded from standard texts; after all, Cartier's proof is quite simple. | |
| Feb 4, 2010 at 14:08 | history | edited | Clark Barwick | CC BY-SA 2.5 |
reorganized on the recommendation of Pete Clark
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| Feb 4, 2010 at 5:54 | comment | added | Pete L. Clark | Thanks again, Clark. I have voted up and accepted your answer, but to me the main point of the answer is the "No, such examples do not exist over a domain" part, so I think it would be even better if that were displayed more prominently. (Note that, as I said above, wikipedia provides an example with $R$ a Boolean ring.) Do you by any chance want to take a shot at the question of whether any of the standard references contain this fact, and if not why not? | |
| Feb 4, 2010 at 5:51 | vote | accept | Pete L. Clark | ||
| Feb 4, 2010 at 5:42 | history | edited | Clark Barwick | CC BY-SA 2.5 |
added 241 characters in body
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| Feb 4, 2010 at 5:05 | history | edited | Clark Barwick | CC BY-SA 2.5 |
clarified the structure of R
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| Feb 4, 2010 at 4:39 | history | edited | Clark Barwick | CC BY-SA 2.5 |
fixed notation, spelling
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| Feb 4, 2010 at 4:19 | history | answered | Clark Barwick | CC BY-SA 2.5 |