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Action of an automorphism of order p on cohomology groups of an algebraic curve. (English) Zbl 0607.14022
Let \(f: X\to Y\) be a finite Galois covering of connected complete non- singular curves over an algebraically closed field k. Let \(G=Gal(X/Y)\), and let \({\mathcal F}\) be a locally free G-sheaf on X. The author [Invent. Math. 75, 1-8 (1984) and J. Number Theory 22, 115-123 (1986; Zbl 0602.14017)] has determined the structure of \(H^ i(X,{\mathcal F})\) as a k[G]-module when f is tamely ramified.
In this article, the case where G is a cyclic group of order \(p=char k>0\) is considered for an invertible G-sheaf \({\mathcal L}\) on X satisfying deg \({\mathcal L}>2g_ X-2\) with \(g_ X=genus\quad of\quad X.\) Under these conditions, the k[G]-module structure of \(H^ 0(X,{\mathcal L})\) is determined. - A characterization of tamely ramified Galois coverings in terms of the projectivity of certain \(H^ 0(X,{\mathcal L})\) is also established. In addition, the author examines the problem of lifting the curve X together with an automorphism \(\sigma\) of order \(p=char k>0\) to the Witt vector ring W(k). It is shown that if \(p\geq 5\) and \(g_ X\geq 2\), then the pair (X,\(\sigma)\) has no lifting to W(k) if \(\sigma\) fixes at least one point on X. - As a corollary, if \(p\geq 5\) and \(g_ X\geq 2\), then a Galois covering \(f: X\to Y\) does not lift to W(k) if f is wildly ramified.
Reviewer: L.D.Olson

14H30 Coverings of curves, fundamental group
14L30 Group actions on varieties or schemes (quotients)
14F25 Classical real and complex (co)homology in algebraic geometry
Full Text: DOI
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