×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Curtis, C.W.; Reiner, I., ()
[2] Deligne, P.; Mumford, D., The irreducibility of the space of curves of given genus, Publ. IHES, 36, 75-109, (1969) · Zbl 0181.48803
[3] Grothendieck, A., Revêtements etales et groupe fondamental (SGA 1), () · Zbl 1039.14001
[4] Hartshorne, R., Algebraic geometry, (1977), Springer Berlin · Zbl 0367.14001
[5] Hasse, H., Theorie der relativ-zyklischen algebraischen funktionenkörper, insbesondere bei endlichem konstantenkörper, J. reine u. angew. math., 172, 37-54, (1934) · JFM 60.0097.01
[6] (=H. Hasse, Math. Abh. Band 2, 133-150).
[7] Heller, A.; Reiner, I., Representations of cyclic groups in rings of integers, I. ann. math., 76, 2, 73-92, (1962) · Zbl 0108.03101
[8] Nakajima, S., On Galois module structure of the cohomology groups of an algebraic variety, Invent. math., 75, 1-8, (1984) · Zbl 0616.14007
[9] Nakajima, S., Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties, J. number theory, 22, 115-123, (1986) · Zbl 0602.14017
[10] F. Oort and T. Sekiguchi, The canonical lifting of an ordinary Jacobian variety need not be a Jacobian variety, Preprint. · Zbl 0605.14031
[11] T. Sekiguchi and F. Oort, On the deformation of Artin-Schreier to Kummer, Preprint. · Zbl 0714.14024
[12] Serre, J.-P., Corps locaux, (1968), Hermann Paris
[13] Valentini, R.C.; Madan, M.L., Automorphisms and holomorphic differentials in characteristic p, J. number theory, 13, 106-115, (1981) · Zbl 0468.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.