Timeline for vanishing of étale cohomology groups with small support with values in an abelian scheme
Current License: CC BY-SA 3.0
11 events
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| Feb 2, 2012 at 20:28 | vote | accept | CommunityBot | moved from User.Id=19475 by developer User.Id=69903 | |
| Feb 1, 2012 at 19:20 | answer | added | user19475 | timeline score: 0 | |
| Dec 2, 2011 at 12:44 | comment | added | user19475 | Does anyone have a vague idea how to approach 3? | |
| Dec 1, 2011 at 11:04 | comment | added | naf | The paper of Murre you refer to is indeed the one I meant. The result of Abhyankar I was referring to is Theorem 1.2 in VI.1 of Kollar's book "Rational curves...". I now see that Abhyankar's theorem actually predates Murre's and it suffices for what I said. | |
| Dec 1, 2011 at 10:30 | history | edited | user19475 | CC BY-SA 3.0 |
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| Dec 1, 2011 at 10:12 | comment | added | user19475 | Dear ulrich, thank you for your response. By the theorem of Murre, do you mean the article "On a connectedness theorem ..."? And which paper of Abhyankar do you mean? | |
| Nov 25, 2011 at 15:27 | comment | added | naf | 2. is also true if the order of $X \in H^1(S\backslash Z,A)$ is prime to the characteristic. If this holds then $X$ comes from an element of $H^1(S \backslash Z,A[n])$ for some $n$ prime to the characteristic . As you suggested, we can use purity, actually just the classical Zariski-Nagata purity theorem, to extend this to an element of $H^1(S,A[n])$ and then we get the desired element of $H^1(S,A)$. | |
| Nov 25, 2011 at 13:03 | comment | added | naf | 1. is true because $S$ is smooth. The trivialisation over $S \backslash Z$ gives a rational map from $S$ to the principal homogenous space and any such map (with $S$ a regular scheme) extends to a morphism. The reason for this is that abelian varieties do not contain rational curves but all positive dimensional fibres of a birational proper morphism $S' \to S$ are covered by rational curves (by a theorem of Murre or Abhyankar (depending on the level of generality)). | |
| Nov 25, 2011 at 12:43 | history | edited | user19475 | CC BY-SA 3.0 |
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| Nov 25, 2011 at 12:32 | history | edited | user19475 | CC BY-SA 3.0 |
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| Nov 23, 2011 at 14:12 | history | asked | user19475 | CC BY-SA 3.0 |