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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)
##### Citations:
Zbl 0369.12011; Zbl 0429.14011
Full Text:
##### References:
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