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Étale \(p\)-covers in characteristic \(p\). (English) Zbl 0558.14009

The main result of the paper, of which three proofs are given (two of them due to L. Moret-Bailly and N. M. Katz, respectively) is the following theorem: Suppose that \(X\) is a separated scheme of finite type over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(G\) be a finite \(p\)-group acting freely on \(X\) and \(Y=X/G\). Then \(\chi_ c(X,{\mathbb Q}_ p)=| G| \cdot \chi_ c(Y,{\mathbb Q}_ p).\) Here \(\chi_ c\) denote the Euler-Poincaré characteristic for cohomology with compact supports.
There are some consequences of which the following three should be mentioned:
(1) the formula of Deuring-Shafarevich for the \(p\)-ranks of \(X\) and \(Y\) for a finite morphism \(f: X\to Y\) of smooth projective curves over \(k\) [cf. e.g. M. L. Madan, Manuscr. Math. 23, 91–102 (1977; Zbl 0369.12011)];
(2) the fact that \(\pi_ 1(X)\) has no p-torsion [in the case of the algebraic closure of a finite field this was proved by T. Katsura, C. R. Acad. Sci., Paris, Sér. A 288, 45–47 (1979; Zbl 0429.14011)]; and
(3) if \(p=2\) and \(X\) is a singular Enriques surface, then its double-cover \(K3\)-surface is ordinary.
Reviewer: H.Lange

MSC:

14E20 Coverings in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
14F35 Homotopy theory and fundamental groups in algebraic geometry
14G15 Finite ground fields in algebraic geometry
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14L30 Group actions on varieties or schemes (quotients)
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References:

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