Timeline for Exercise concerning locally constant presheaves [closed]
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 11, 2011 at 6:45 | comment | added | Pete L. Clark | @Matt: I agree, I missed that $V$ is not in $\mathcal{U}$. Thanks for pointing that out. | |
| May 11, 2011 at 5:33 | comment | added | Emerton | ... together with the stalkwise criterion for isomorphisms of sheaves will help); and (3) is also standard, but I'm not sure whether it will be in Spanier --- its probably the trickiest of the three points if you aren't practiced with sheaf arguments. Once you have these, (1) and (3) and your initial assumption will show that your composite map is surjective as well as injective, QED. Regards, Matthew | |
| May 11, 2011 at 5:30 | comment | added | Emerton | Dear Jesko, Use the following facts: (1) sheafification doesn't change stalks. (2) The sheafification of a locally constant presheaf (as you have defined it) is a locally constant sheaf (as it is usually defined). (3) If $\mathcal F^+$ is a locally constant sheaf (e.g. the sheaf obtained by sheafifying your given locally constant presehaf!) then the natural map $\mathcal F^+(V) \to \mathcal F_P$ is injective for any connected open set. Of these facts, (1) is standard --- and probably in Spanier; (2) is not so bad, and may also be in Spanier (if you want to prove it yourself, point (1) ... | |
| May 11, 2011 at 5:20 | comment | added | Emerton | Dear Pete, Since $V$ is not necessarily in $\mathcal U$, and since $\mathcal F$ is just a presheaf, it seems to me that $\mathcal F(V)$ could be essentially arbitrary, and hence the first map need not be an isomorphism. | |
| May 11, 2011 at 5:02 | comment | added | Pete L. Clark | Note also that saying "$\mathcal{F}(U) = \mathcal{F}_P$" seems slightly sloppy to me: what you mean is that the canonical map from the guy on the left to the guy on the right is an isomorphism. | |
| May 11, 2011 at 5:00 | comment | added | Pete L. Clark | I sympathize with the OP's comment. I think what Scott means is that no matter how much of an expert one is in sheaf theory, the way to solve the question is to write out all the definitions and follow your nose. It's not dead easy, but there's no one place to point to and say "aha". (Except: have you tried using the universal property of sheafification to show that a locally constant sheaf on a connected space is constant? That seems to be the matter of it: if I have it right, both maps are isomorphisms, and that the first one is is obvious; that the second is uses what I just said.) | |
| May 11, 2011 at 4:49 | comment | added | S. Carnahan♦ | The question you linked, while easy, is interesting because it is about local-versus-global phenomena. Your question is a diagram chase. | |
| May 10, 2011 at 23:19 | comment | added | Jesko Hüttenhain | I'm sorry, I had read the FAQ and I am not quite sure how my questions are inappropriate. In particular, how is this particular question any less demanding or interesting than, say, that one: mathoverflow.net/questions/24361/… | |
| May 10, 2011 at 22:48 | comment | added | S. Carnahan♦ | Perhaps you should try asking on math.stackexchange.com | |
| May 10, 2011 at 22:48 | history | closed | S. Carnahan♦ | too localized | |
| May 10, 2011 at 22:44 | comment | added | Martin Brandenburg | I don't think that your recent questions are appropriate for MO. Or at least you should put some effort to solve these questions by your own and show your ideas ... | |
| May 10, 2011 at 22:42 | history | asked | Jesko Hüttenhain | CC BY-SA 3.0 |