Crew, Richard M. Étale \(p\)-covers in characteristic \(p\). (English) Zbl 0558.14009 Compos. Math. 52, 31-45 (1984). 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 Cited in 2 ReviewsCited in 34 Documents 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) Keywords:étale \(p\)-covers; torsionless fundamental group; group acting on scheme; \(p\)-ranks of smooth projective curves; characteristic \(p\); Euler-Poincaré characteristic; singular Enriques surface Citations:Zbl 0369.12011; Zbl 0429.14011 PDFBibTeX XMLCite \textit{R. M. Crew}, Compos. Math. 52, 31--45 (1984; Zbl 0558.14009) Full Text: Numdam EuDML References: [1] M. Artin and D. Mumford : Some elementary examples of unirational varities which are not rational . Proc., London Math. Soc. 25, pp. 75-95. · Zbl 0244.14017 [2] P. 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