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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


The étale cohomology of $ p$-torsion sheaves. I

Author: William Anthony Hawkins
Journal: Trans. Amer. Math. Soc. 301 (1987), 163-188
MSC: Primary 14F20; Secondary 14L15
MathSciNet review: 879568
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Abstract: This paper generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincaré characteristic of a constructible sheaf of $ {F_l}$-modules on a smooth, proper curve, over an algebraically closed field $ k$ of characteristic $ p > 0$, as a sum of local and global terms, where $ l \ne p$. The primary focus is on removing the restriction on $ l$. We begin with calculations for $ p$-torsion sheaves trivialized by $ p$-extensions, but using etale cohomology to give a unified proof for all primes $ l$.

In the remainder of this work, only $ p$-torsion sheaves are considered. We show the existence on $ {X_{{\text{et}}}}$, $ X$ a scheme of characteristic $ p$, of a short exact sequence of sheaves, involving the tangent space at the identity of a finite, flat, height 1, commutative group scheme, and the subsheaf fixed by the $ p$th power endomorphism; the latter turns out to be an etale group scheme. A corollary gives complete results on the Euler-Poincaré characteristic of a constructible sheaf of $ {F_p}$-modules on a smooth, proper curve, over an algebraically closed field $ k$ of characteristic $ p > 0$, when the generic stalk has rank $ p$.

Explicit computations are given for the Euler characteristics of such $ p$-torsion sheaves on $ {P^1}$ and a result on elliptic surfaces is included. A study is made of the comparison of the $ p$-ranks of abelian extensions of curves. Several examples of $ p$-ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of $ {F_q}$-modules, $ q={p^r},\,r \ge 1$.

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  • [1] N. Bourbaki, Éléments de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie, Actualités Sci. Ind. No. 1285. Hermann, Paris, 1960 (French). MR 0132805 (24 #A2641)
  • [2] Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local\ par Michiel Hazewinkel. MR 0302656 (46 #1800)
  • [3] Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). MR 0274458 (43 #223a)
  • [4] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
  • [5] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
  • [6] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985 (44 #219)
  • [7] F. Oort, Commutative group schemes, Lecture Notes in Mathematics, vol. 15, Springer-Verlag, Berlin-New York, 1966. MR 0213365 (35 #4229)
  • [8] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237 (82e:12016)
  • [9] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380 (56 #8675)
  • [10] William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117 (82e:14003)
  • [11] Edwin Weiss, Algebraic number theory, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR 0159805 (28 #3021)
  • [12] Helmut Hasse, Zyklische unverzweigte Erweiterungskörper vom Primzahlgrade 𝑝 über einem algebraischen Funktionenkörper der Charakteristik 𝑝, Monatsh. Math. Phys. 43 (1936), no. 1, 477–492 (German). MR 1550551,
  • [13] Manohar L. Madan, On a theorem of M. Deuring and I. R. Šafarevič, Manuscripta Math. 23 (1977/78), no. 1, 91–102. MR 0460335 (57 #329)
  • [14] Yu. I. Manin, The Hasse-Witt matrix of an algebraic curve, Amer. Math. Soc. Transl. (2) 45 (1965), 245-264.
  • [15] M. Raynaud, Caracteristique d'Euler-Poincare d'un faisceau et cohomologie des varietes abeliennes, (Seminaire Bourbaki 1964/1965, no. 286); also in Dix Exposes sur la Cohomologie des Schemas, North-Holland, Amsterdam, 1968, pp. 12-30.
  • [16] Jean-Pierre Serre, Sur la topologie des variétés algébriques en caractéristique 𝑝, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24–53 (French). MR 0098097 (20 #4559)
  • [17] Doré Subrao, The 𝑝-rank of Artin-Schreier curves, Manuscripta Math. 16 (1975), no. 2, 169–193. MR 0376693 (51 #12868)
  • [18] John Tate and Frans Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4) 3 (1970), 1–21. MR 0265368 (42 #278)

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PII: S 0002-9947(1987)0879568-5
Article copyright: © Copyright 1987 American Mathematical Society

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