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I remember reading that if $X/\mathbf F_q$ is a projective smooth global complete intersection, then the characteristic polynomial of the $\mathbf F_q$-linear Frobenius of $X$ on $H^i_{et}(X\otimes\overline{\mathbf F_q},\mathbf Q_l)$, $l\nmid q$, has integer coefficients and is independent of $l$.

How does one prove this fact?


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By being Deligne? You might want to take a look at Deligne's Weil I here: archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1974__43_/… Especially Theorem 1.6. –  Keerthi Madapusi Pera Jun 16 '11 at 19:43
You may read Deligne's Weil I for the case of projective smooth varieties (not necessarily complete intersection), or certain complete intersections before Weil I, again by Deligne. See also Katz's two articles in Motives I and Illusie's Miscellany for what are known and what are unknown. If you want an elementary proof for the indep. of $l$ of only the Betti numbers in the case of complete intersections, note that they are liftable to char. 0. –  shenghao Jun 16 '11 at 19:51
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2 Answers

up vote 2 down vote accepted

If you allow a proof using the Weil conjectures, there's an otherwise simple proof of this for any smooth projective variety in Deligne's first paper proving the Weil conjectures. See pages 276-277 of Weil I.

Edit: Sorry Keerthi! Didn't see your comment when I posted this.

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As was pointed above, this statement was proven by Deligne for general smooth projective varieties. Yet in your case you don't require the theory of Deligne's weights, since you have only one cohomology group (the middle one) that could vary. Using the latter statement, you can easily deduce your fact from the properties of etale cohomology proved by Grothendieck and Artin.

You can probably find more detail here: http://www.jmilne.org/math/CourseNotes/LEC.pdf

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