Timeline for Flatness and local freeness
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2015 at 19:05 | comment | added | Daniel Litt | @user26857: Added a link to the original example in Ravi's notes--thanks! | |
| Dec 31, 2015 at 19:04 | history | edited | Daniel Litt | CC BY-SA 3.0 |
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| Dec 31, 2015 at 18:46 | comment | added | user26857 | The link to the example is dead. (It can be replaced by a reference: Lam, Exercises in Modules and Rings, Exercise 4.17. Also here.) | |
| Jul 21, 2011 at 20:49 | comment | added | Daniel Litt | (The link above should be math.stanford.edu/~vakil/216blog ...I think MO screwed up the html.) | |
| Jul 21, 2011 at 17:51 | comment | added | Ravi Vakil | Also, that example in your UPDATE is great. Does anyone know a published reference with an argument? (I now mention it as an aside in Exercise 25.4.E in the July ~21 notes <a href="math.stanford.edu/~vakil/216blog/">here</a>.) | |
| Jul 20, 2011 at 17:08 | comment | added | Daniel Litt | Yeah, this is kind of an amazing fact :). | |
| Jul 20, 2011 at 16:36 | comment | added | Ravi Vakil | @Daniel --- as I now mention in my "answer", finite presentation can be weakened to finitely generated. (Your argument is the slickest one I know in the finitely presented case though.) | |
| Jul 27, 2010 at 16:49 | history | edited | Daniel Litt | CC BY-SA 2.5 |
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| Jul 27, 2010 at 16:02 | history | edited | Daniel Litt | CC BY-SA 2.5 |
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| Jul 27, 2010 at 15:46 | history | edited | Daniel Litt | CC BY-SA 2.5 |
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| Jul 27, 2010 at 15:46 | comment | added | Keenan Kidwell | Okay, yeah, the Tor sequence. Thanks. | |
| Jul 27, 2010 at 15:41 | history | edited | Daniel Litt | CC BY-SA 2.5 |
Added counterexample.
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| Jul 27, 2010 at 15:37 | comment | added | Daniel Litt | @Keenan: The exactness of that sequence is exactly where I'm using the flatness of $M$. Consider the LES of Tor. And finite flat over a Noetherian local ring implies free, so obviously projective. | |
| Jul 27, 2010 at 15:33 | comment | added | Keenan Kidwell | That should be "Noetherian local ring." | |
| Jul 27, 2010 at 15:32 | comment | added | Keenan Kidwell | Aren't you saying that the sequence obtained $0\rightarrow K\rightarrow A^n\rightarrow M\rightarrow 0$ by lifting a basis for $M/\mathfrak{m}M$ remains exact upon tensoring with the residue field? I don't understand why this is the case, maybe 'cause I don't see where you're using the flatness of $M$. Is it true that $M$ is necessarily projective, i.e., does finite flat over a Noetherian ring imply projective? | |
| Jul 27, 2010 at 15:26 | comment | added | ashpool | I agree with Akhil's answer wholeheartedly. But the question is whether we can relax the condition of finite presentation with finite generation. | |
| Jul 27, 2010 at 15:17 | history | answered | Daniel Litt | CC BY-SA 2.5 |