Timeline for Euler characteristic of local system depends only on rank?
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9 events
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| Feb 7, 2017 at 22:50 | comment | added | Will Sawin | @user84144 One views the cohomology of a local system coming from a representation $\pi$ of a finite quotient $G$ of the fundamental group as the space of homomorphisms from $\pi^\vee$ to the cohomology of the corresponding $G$-cover. Then apply character theory to calculate the dimension of this in terms of traces of automorphisms. I guess one has to compare the mod-$\ell$ Homs to some $\ell$-adic Homs and use character theory there. This is all in the paper of Illusie, it's just that there is a bunch of additional stuff in that paper to gable ramification that is not needed here. | |
| Feb 7, 2017 at 20:57 | comment | added | user84144 | @WillSawin, can you elaborate on your last two sentences? What is the trace formula expressing euler characteristic of a local system as sum of traces of automorphisms? | |
| Nov 9, 2016 at 1:27 | comment | added | Will Sawin | @JohnPardon The Lefschetz fixed point formula requires the variety to be proper. | |
| Nov 8, 2016 at 20:25 | comment | added | John Pardon | @WillSawin Great, just let me ask a stupid question: at which point in that argument is properness used? | |
| Nov 8, 2016 at 4:37 | comment | added | Will Sawin | @JohnPardon If I recall correctly the argument of that paper is essentially due to Deligne but was written up by Illusie. However even the result of Deligne and Illusie is more general than the situation requires. This follows immediately from the Lefschetz fixed point formula of SGA applied to a finite etale cover on which the bundle trivializes. The Euler characteristic may be computed as a sum of traces of the automorphisms of the bundle. The nontrivial automorphisms have no fixed points, so no contribution, and the contribution of the trivial automorphism depends only on the rank. | |
| Nov 7, 2016 at 18:15 | comment | added | John Pardon | Kato--Saito cites Illusie in the proof of Theorem 4.2.9. Reading the MR review, it would seem that the answer to my question is actually due to Illusie: ams.org/mathscinet-getitem?mr=629127 | |
| Nov 7, 2016 at 18:11 | comment | added | John Pardon | Thanks! As stated, the theorem you cite only applies when $X$ is smooth. Any idea whether the methods would generalize to the non-smooth case? | |
| Nov 7, 2016 at 18:09 | vote | accept | John Pardon | ||
| Nov 6, 2016 at 17:52 | history | answered | js21 | CC BY-SA 3.0 |