×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
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 · doi:10.1112/plms/s3-25.1.75
[2] P. Berthelot : Cohomologie cristalline des Schemas de caracteristique p > 0 . Lecture Notes in Math. #407, Springer 1974. · Zbl 0298.14012
[3] P. Berthelot and A. Ogus : Notes on crystalline cohomology . Mathematical Notes #21, Princeton University Press, 1978. · Zbl 0383.14010 · doi:10.1515/9781400867318
[4] E. Bombieri and D. Mumford : Enriques classification of surfaces in char. p. III . Inv. Math. 35 (1976) 197-232. · Zbl 0336.14010 · doi:10.1007/BF01390138 · eudml:142405
[5] P. Blass : Unirationality of Enriques surfaces in characteristic two , (to appear). · Zbl 0549.14019 · numdam:CM_1982__45_3_393_0 · eudml:89543
[6] L. Illusie : Complexe de De Rham-Witt et cohomologie cristalline . Ann. Scient. Ec. Norm. Sup. 4e serie t. 12 (1979) 501-661. · Zbl 0436.14007 · doi:10.24033/asens.1374 · numdam:ASENS_1979_4_12_4_501_0 · eudml:82043
[7] T. Katsura : Surfaces unirationelles en caracteristique p . C.R. Acad. Sc. Paris, t. 288 series A (1979) 45-47. · Zbl 0429.14011
[8] N. Katz : Travaux de Dwork . In: Sem Bourbaki, exp. 409 . Lecture Notes in Math. #317, Springer 1973. · Zbl 0259.14007 · numdam:SB_1971-1972__14__167_0 · eudml:109810
[9] N. Nygaard : On the fundamental group of a unirational 3-fold . Inv. Math. 44 (1978) 75-86. · Zbl 0427.14014 · doi:10.1007/BF01389903 · eudml:142520
[10] J.-P. Serre : On the fundamental group of a unirational variety . J. London Math. Soc. 1(14) (1959) 481-484. · Zbl 0097.36301 · doi:10.1112/jlms/s1-34.4.481
[11] J.-P. Serre : Representations Lineaires des Groupes Finis , 2nd edn., Paris, Hermann 1971. · Zbl 0223.20003
[12] I. Shafarevich : On p-extensions . Mat. Sbornik 20 (1947) 351-363 (AMS Translations Ser. 2 vol. 4, 1956). · Zbl 0041.17101
[13] S. Shatz : Profinite groups, arithmetic, and geometry . Ann. Math. Studies #67, Princeton University Press 1972. · Zbl 0236.12002 · doi:10.1515/9781400881857
[14] T. Shioda : An example of unirational surfaces in characteristic p . Math. Ann. 211 (1974) 233-236. · Zbl 0276.14018 · doi:10.1007/BF01350715 · eudml:162649
[15] A. Grothendieck : Revêtements etales et groupe fondamental . Lecture Notes in Math 224, Springer-Verlag 1971. · Zbl 0234.14002 · doi:10.1007/BFb0058656 · arxiv:math/0206203
[16] M. Artin , A. Grothendieck and J.-L. Verdier : Théorie des topos et cohomologie etale des schémas . Lecture Notes in Math. 269, 270, 305, Springer-Verlag 1972 -1973. · Zbl 0245.00002 · doi:10.1007/BFb0070714
[17] P. Deligne : Cohomologie etale . Lecture Notes in Math. 569, Springer-Verlag 1977. · Zbl 0345.00010 · doi:10.1007/BFb0091516
[18] A. Grothendieck : Cohomologie l-adique et fonctions L . Lecture Notes in Math. 589, Springer-Verlag 1977. · Zbl 0345.00011 · doi:10.1007/BFb0096802
[19] P. Berthelot , A. Grothendieck and L. Illusie : Théorie des intersections et théorème de Riemann-Roch . Lecture Notes in Math. 225, Springer-Verlag 1971. · Zbl 0218.14001
[20] A. Grothendieck , M. Raynaud , D. Rim , P. Deligne and N. Katz : Groupes de Monodromie en Géométrie Algébrique . Lecture Notes in Math. 288-340, Springer-Verlag 1972-1973.
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.