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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

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