Timeline for Reference wanted - etale sheaves on $X$ versus on $\overline{X}$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2012 at 9:13 | vote | accept | Sasha | ||
| Feb 9, 2012 at 19:00 | answer | added | user19475 | timeline score: 2 | |
| Feb 9, 2012 at 17:15 | comment | added | Sasha | @Dror: I don't know details, but my general feeling says that your argument does not contradict my statement? In my understanding, Serre-Hohcschild follows from interpretation of the unstructured global sections $Sh_G (Y) \to Ab$ as composition of $\Gamma: Sh_G (Y) \to G-mod$ with invariants $inv: G-mod \to Ab$; Where usually we already understand that $Sh_G (Y)$ is equivalent to $Sh(G\Y)$, for the $G\Y$ we have in mind. I might be wrong. | |
| Feb 9, 2012 at 16:34 | comment | added | Dror Speiser | Hey Sasha! I don't think this has to be an equivalence of categories. If $X$ is a smooth projective variety, then the Hochschild-Serre spectral sequence gives us the sequence: $0\rightarrow Pic(X)\rightarrow Pic(\bar{X})^\Gamma\rightarrow Br(k)$, where it is known that the last map is not always zero. Using that $Pic$ describes invertible sheaves, it seems that sometimes the category $Sh(\bar{X}_{et})^\Gamma$ is bigger. A similar situation occurs when you go up one notch and consider $O_X$-algebras. There you can have equivariant Azumaya algebras that are not base-change. | |
| Feb 9, 2012 at 13:47 | history | asked | Sasha | CC BY-SA 3.0 |