Timeline for Do isogenies between AVs over finite fields separate finite subgroups?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 2, 2011 at 16:29 | vote | accept | Tommaso Centeleghe | ||
| Mar 2, 2011 at 16:29 | comment | added | Tommaso Centeleghe | You are right. Now I understand, I answered my own question with the proof of Waterhouse's original claim. | |
| Mar 2, 2011 at 14:31 | comment | added | Yuri Zarhin | OOPS! Sorry. In the last sentence one should read $J\subset End_k(A)$. | |
| Mar 2, 2011 at 14:16 | comment | added | Yuri Zarhin | You are welcome. Well, units in $End_k(B)$ are actually automorphisms of $B$, so they allow us to vary $\lambda$'s while $\ker(\lambda)$ remains the same. As for Waterhouse actual claim, please notice that at this point he deals not with arbitrary finite group subschemes of $A$ but only with those that (slightly reformulating) are kernels of homomorphisms $A \to A^m$. He does not claim that every finite group subscheme $H \subset A$ coincides with $H(J)$ for a certain ideal $J \subset \End_k(B)$. | |
| Mar 2, 2011 at 14:03 | comment | added | Tommaso Centeleghe | Thanks, this helps and, together with Chris' example, convince me that what I was asking cannot be true. Let me just point out that your endomorphism ring End_k(B) better do not have too many units, for otherwise you may be able to pick several generators for your principal ideal, whose associated kernels need not coincide. right? What about Waterhouse's actual claim, posted in my EDIT2 above? Can you sketch a proof of it? (I realized that what it says is actually weaker than what I was asking) | |
| Mar 2, 2011 at 14:02 | history | edited | Yuri Zarhin | CC BY-SA 2.5 |
typos corrected
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| Mar 2, 2011 at 13:54 | history | answered | Yuri Zarhin | CC BY-SA 2.5 |