Timeline for Picard groups of non-projective varieties
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Nov 12 at 9:47 | comment | added | Doug Liu | Dear @Lars, Why "Representability as an algebraic space is probably too much to ask for"? | |
| Aug 5, 2010 at 16:30 | answer | added | Simon Pepin Lehalleur | timeline score: 6 | |
| Aug 3, 2010 at 13:16 | comment | added | Lars |
If I am not mistaken, then for a proper surjective hypercovering, one also has $H^1(U_{\bullet},\mu_{n. U_\bullet})\cong H^1(U,\mu_{n,U})$, so we get control over the prime-to-$p$-torsion, by considering the exponential sequence $\mu_{n,U_\bullet}\rightarrow \mathcal{O}_{U_\bullet}^\times \rightarrow \mathcal{O}_{U_\bullet}$ on $U_\bullet$, and $Pic(U_\bullet)\cong H^1(U_\bullet,\mathcal{O}_{U_\bullet})$.
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| Jul 30, 2010 at 19:26 | comment | added | Tony Scholl | If $f:U'\to U$ is an alteration with $U$, $U'$ both smooth then $f$ is flat outside some $Z\subset U$ of codimension $\ge2$, so you can verify triviality of $L$ on the complement, where there is faithfully flat descent. So if you do the usual simplicial thing with successive alterations you should get $Pic(U)$ injecting into $Pic(U_\bullet)$. | |
| Jul 30, 2010 at 14:56 | comment | added | Lars | Thanks Brian! One thing I would like to be able to do is, for a line bunde L on U (smooth quasi-projective), to determine whether L is trivial by pulling back to some suitable alteration, or proper hypercovering of U. Also, do you have a reference for non-proper simplicial methods? | |
| Jul 30, 2010 at 10:34 | comment | added | BCnrd | Dear Lars: For what you want, perhaps some non-proper simplicial methods can be used? (Not quite sure what kind of examples or applications you have in mind with your question, so this suggestion is a bit vague.) One place to look is at the computation of the Picard groups of some Artin stacks. | |
| Jul 30, 2010 at 8:10 | comment | added | Lars | Hehe, they come precisely from browsing through your notes on cohomological descent and from the "general concept" (which apparently is none) that cohomology can be computed via surjective proper hypercoverings. By "naive" I guess I ment "uninformed". I've also seen "simplicial Picard" groups mentioned in a few places and I guess I was hoping for comparison theorems for the simplicial Picard group of a hypercovering of X with the Picard group of X | |
| Jul 30, 2010 at 3:10 | comment | added | BCnrd | Where does your "naive impression" related to proper hypercovers come from? It sounds dubious. Proper hypercovers are good for descent with abelian etale sheaves due to the proper base change theorem, but when you bring in line bundles over the structure sheaf you enter the world of quasi-coherent sheaves, for which it doesn't seem like proper hypercoverings (without some kind of flatness conditions which seem impractical) will be useful -- but I'd be happy to hear to the contrary. My recollection is that nothing much can be said without properness conditions on $X$ over $k$. | |
| Jul 29, 2010 at 18:01 | history | edited | Lars | CC BY-SA 2.5 |
added 91 characters in body
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| Jul 29, 2010 at 17:33 | history | asked | Lars | CC BY-SA 2.5 |