Timeline for Lemma in Scholze-Weinstein
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2015 at 22:25 | comment | added | Toby | I think I see now. If $\{a_i \} \in \lim K/\Lambda_i$, then $\{a_i + \Lambda_i\}$ is a descending sequence of closed unit discs. If $K$ is spherically complete, then there is some $a \in \cap_i a_i + \Lambda_i$, which means exactly that $a \in K$ maps to $\{a_i\}$ in $\lim K/\Lambda_i$. Thanks you. | |
| Dec 31, 2015 at 22:02 | comment | added | nfdc23 | It sheds a lot of light on the present situation (and in fact explains everything): think amount the geometric meaning of surjectivity of $K \rightarrow \lim K/\Lambda_i$ and then stare at the definition of spherical completeness. | |
| Dec 31, 2015 at 18:55 | comment | added | Toby | Sorry for being slow. My proof of the exercise in Weibel's book is to use the short exact sequence $0 \to \{A_i\} \to \{A\} \to \{A/A_i\} \to 0$, apply derived functors, observe that the middle system has a vanishing $\lim^1$ as its constant, and thus arrive at: $A \to \lim A/A_i$ is surjective (i.e., each Cauch sequence has a limit) iff $\lim^1 A_i = 0$. This doesn't seem to help shed light on the present situation though. | |
| Dec 31, 2015 at 17:30 | comment | added | nfdc23 | Your use of spherical completeness is too limited (why apply to that single descending sequence of discs, which anyway all contain 0!); it really gives you much more. Please do the exercise in Weibel's book, and then you should see the rest. I shouldn't have made the statement about "bounded away from 0" for radii; it isn't used here (but highlights the relevance of the comment about possibly non-unique limits near the end of the exercise in Weibel's book). | |
| Dec 31, 2015 at 17:09 | comment | added | Toby | @ndfc23: thanks, I edited the question to add the missing hypothesis. If I literally applied the spherical completeness hypothesis to the descending sequence $\{ \Lambda_n \}$ of closed discs, then, I think, I would simply learn that $\cap \Lambda_n \neq 0$. This does not use the information that the radii are bounded away from $0$, so I'm guessing you had something smarter in mind. Would you mind elaborating on it? Thanks in advance. | |
| Dec 31, 2015 at 17:05 | history | edited | Toby | CC BY-SA 3.0 |
added hypothesis on ideals
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| Dec 31, 2015 at 9:48 | comment | added | nfdc23 | @Corbennick: The OP is asking about the final assertion in the proof of Lemma 5.2.7 in that paper, a step which is left to the reader (and so not proved in the paper), upon imposing the omitted condition to which you allude (i.e., each nonzero ideal $\Lambda_i$ is finitely generated, or equivalently is a "closed disk"). Exercise 3.5.1 in Weibel's book on homological algebra is a good reference for Toby to see that the requested vanishing is literally a restatement of the spherical completeness of $K$ (using that the $\Lambda_i$'s are closed discs with radii bounded away from 0). | |
| Dec 31, 2015 at 7:24 | comment | added | user1688 | First you forgot a condition, second: there IS given a proof in that paper! | |
| Dec 31, 2015 at 6:28 | review | First posts | |||
| Dec 31, 2015 at 9:14 | |||||
| Dec 31, 2015 at 6:26 | history | asked | Toby | CC BY-SA 3.0 |