Timeline for The fiber of the sheaf of invariants
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2010 at 17:43 | comment | added | Rurik | I see... So, as I feared, I did get it completely wrong :-(... thank you again and sorry for taking so much of your time! best regards | |
| Jul 14, 2010 at 13:55 | comment | added | t3suji | Take a point $y\in Y$. The map $\pi:X\to Y$ is finite; consider the scheme-theoretic preimage $\pi^{-1}(x)$. For any $G$-equivariant sheaf $F$ on $X$, the following spaces are identified: - The fiber of the sheaf of invariants $\pi^*(F)^G$ at $y$; - The $G$-invariants in the fiber $\pi_*F(y)$ of $\pi_*F$ at $y$; - The $G$-invariants in the sections $H^0(\pi^{-1}(y),F)$. For the last description, you must take the scheme-theoretic preimage. For instance, even if $\pi^{-1}(y)$ is a single point $x$ with non-trivial scheme structure (which IIUC is what you ask), it is not just $F(x)^G$. | |
| Jul 14, 2010 at 10:10 | comment | added | Rurik | I am sorry to bother you again, but there are some things I still do not get: if X is an abelian variety and Y its (singular) kummer variety (char$k\neq 2$) then the action is not free and the quotient map is not flat. If I have a G−sheaf on $X$ then I canconsider the sheaf $\pi_*F^G$ If I take the fiber at a a singular point $\overline{x}$ what do I obtain? I believed to get $\pi_*(F(x))^G$ but I do not think this is the same as $(\pi_*(F)(\overline{x}))^G$ (that is quoting t3suji the 'invariants in the fiber of the direct image'). What I am doing wrong? Thanks again and again sorry! | |
| Jul 9, 2010 at 14:06 | comment | added | Boyarsky | Dear Stgermain: "unit on a scheme" means "non-vanishing section of the structure sheaf". So an integer $n$ is a unit on a scheme $S$ when char($k(s)$) doesn't divide $n$ for all $s \in S$, or equivalently $S$ is a $\mathbb{Z}[1/n]$-scheme. As for "equivariant culture", by which I assume you mean "general stuff about quotients by group actions", begin by learning about smooth & etale maps (Milne's book on etale cohomology, Chapter 2 of "Neron Models", SGA1...) and then descent theory (early part of Chapter 6 of "Neron Models", SGA1, FGA Explained...) and then read Expose V in SGA3. | |
| Jul 8, 2010 at 15:05 | history | edited | Boyarsky | CC BY-SA 2.5 |
added 374 characters in body
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| Jul 8, 2010 at 14:34 | vote | accept | Rurik | ||
| Jul 8, 2010 at 14:11 | comment | added | Rurik | Wow! Thank you a lot for your answers/comments! That was really helpful. Now I will try to do my homework and proving the case in which the action is not free.. thank you again best wishes Stgermain | |
| Jul 8, 2010 at 12:58 | comment | added | t3suji | On the other hand, one question being asked is simply `is the fiber of the sheaf of invariants equal to the invariants in the fiber of the direct image'? In this form, the answer is still positive (assuming char k and card(G) are coprime) even if the action is not free, and the proof is quite easy. | |
| Jul 8, 2010 at 12:16 | history | answered | Boyarsky | CC BY-SA 2.5 |