# global complete intersection and independence of $l$

Hello,

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?

Thanks

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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