# Lefschetz fixed point formula: an 'easy' proof; cohomology with compact support

I have two questions concerning the LFPF (for etale cohomology).

1. Is there an easy 'explanation' of this statements (that could be understood by students)? In particular, I would like to get away with a short and 'simple' list of properties of the cycle class map, and I want to make the linear algebra part of the proof as 'conceptual' as possible.

2. Is there an 'easy' proof of the non-proper version of the formula (that involves cohomology with compact support)?

I do not plan to give a complete proof; I would rather like to tell how to deduce the statement from certain 'basic' (and 'motivic') properties of etale cohomology (possibly, omitting some details).

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What do you mean by the non-proper version ? The LFPF for the Frobenius endomorphism ? As to an 'easy explanation', the paper by Atiyah and Bott on the LFPF outlines an axiomatic approach. –  Damian Rössler Sep 9 '12 at 9:28
I am sorry; which paper of Atiyah and Bott are you speaking about? I definitely do not want to apply any analytic methods. By non-proper version I mean the version for non-proper (and non-projective) varieties; it involves cohomology with compact support. –  Mikhail Bondarko Sep 9 '12 at 9:41
I mean the paper "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications", Annals of Math. (1968). About the non-proper version: I don't think that there is a version of the LFPF in the non-proper situation. Nevertheless, in that situation, in positive characteristic, there is a conjecture of Deligne saying that if a correspondence is sufficiently twisted by the Frobenius, then there is no "contribution from $\infty$" (ie from the boundary of a compactification). This was proved by Pink and Fujiwara. See Fujiwara "Rigid geometry..." Invent. Math. 127 (1997). –  Damian Rössler Sep 9 '12 at 12:50
Do you think that the results of Atiyah and Bott can be applied to the Frobenius operator? Which section could help here? –  Mikhail Bondarko Sep 9 '12 at 14:16