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Géométrie rigide et cohomologie des variétés algébriques de \(caract\acute eristique\quad p\). (Rigid geometry and cohomology of algebraic varieties of \(characteristic\quad p\)). (French) Zbl 0606.14017

The author gives the basic definitions concerning rigid analytic cohomology, a common generalisation of crystalline cohomology (up to isogeny) and Monsky-Washnitzer cohomology. Among other things he introduces rigid cohomology with compact supports and shows that one has the expected exact sequences, hence giving finiteness results for all varieties which can be embedded in a smooth and proper variety with normal crossing complement. He ends by introducing the notion of overconvergent crystals, the proposed local coefficient systems of the theory.
Reviewer: T.Ekedahl

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry

References:

[1] A. ADOLPHSON , S. SPERBER , Twisted Kloosterman sums and p-adic Bessel functions , Amer. Journ. of Math. 106, n^\circ 3, 549-592 ( 1984 ). MR 85k:12002 | Zbl 0552.12010 · Zbl 0552.12010 · doi:10.2307/2374285
[2] P. BERTHELOT , Cohomologie cristalline des schémas de caractéristique p > 0 , Lecture Notes in Math. 407, Springer Verlag ( 1974 ). MR 52 #5676 | Zbl 0298.14012 · Zbl 0298.14012
[3] P. BERTHELOT , Cohomologie rigide et théorie de Dwork: le cas des sommes exponentielles , Astérisque 119-120, 17-49 ( 1984 ). MR 87b:14009 | Zbl 0577.14013 · Zbl 0577.14013
[4] P. BERTHELOT , Cohomologie rigide et cohomologie rigide à support propre , en préparation.
[5] P. BERTHELOT , A. OGUS , Notes on crystalline cohomology , Math. Notes 21, Princeton Univ. Press ( 1978 ). MR 58 #10908 | Zbl 0383.14010 · Zbl 0383.14010
[6] P. BERTHELOT , A. OGUS , F-isocrystals and De Rham cohomology I , Inventiones Math. 72, 159-199 ( 1983 ). MR 85e:14025 | Zbl 0516.14017 · Zbl 0516.14017 · doi:10.1007/BF01389319
[7] B. DWORK , On the zeta function of a hypersurface: II , Annals of Math. 80, n^\circ 2, 227-299 ( 1964 ). MR 32 #5654 | Zbl 0173.48601 · Zbl 0173.48601 · doi:10.1007/BF02684275
[8] B. DWORK , On the zeta function of a hypersurface: III , Annals of Math. 83, n^\circ 3, 457-519 ( 1966 ). MR 35 #194 | Zbl 0173.48601 · Zbl 0173.48601 · doi:10.1007/BF02684275
[9] B. DWORK , On the zeta function of a hypersurface : IV. A deformation theory for singular hypersurfaces , Annals of Math. 90, n^\circ 2, 335-352 ( 1969 ). MR 41 #8423 | Zbl 0213.47402 · Zbl 0213.47402 · doi:10.2307/1970727
[10] B. DWORK , p-adic cycles , Publ. Math. I.H.E.S. 37, 27-115 ( 1969 ). Numdam | MR 45 #3415 | Zbl 0284.14008 · Zbl 0284.14008 · doi:10.1007/BF02684886
[11] B. DWORK , Bessel functions as p-adic functions of the argument , Duke Math. Journ. 41, 711-738 ( 1974 ). Article | MR 52 #8124 | Zbl 0302.14008 · Zbl 0302.14008 · doi:10.1215/S0012-7094-74-04176-3
[12] R. ELKIK , Solutions d’équations à coefficients dans un anneau hensélien , Ann. Scient. Ec. Norm. Sup., 6, n^\circ 4, 553-604 ( 1973 ). Numdam | MR 49 #10692 | Zbl 0327.14001 · Zbl 0327.14001
[13] J. FRESNEL , M. VAN DER PUT , Géométrie analytique rigide et applications , Progress in Math. 18, Birkhäuser ( 1981 ). MR 83g:32001 | Zbl 0479.14015 · Zbl 0479.14015
[14] L. GERRITZEN , M. VAN DER PUT , Schottky groups and Mumford curves , Lecture Notes in Math. 817, Springer Verlag ( 1981 ). MR 82j:10053 | Zbl 0442.14009 · Zbl 0442.14009 · doi:10.1007/BFb0089957
[15] R. HARTSHORNE , On the De Rham cohomology of algebraic varieties , Publ. Math. I.H.E.S. 45, 5-99 ( 1976 ). Numdam | MR 55 #5633 | Zbl 0326.14004 · Zbl 0326.14004 · doi:10.1007/BF02684298
[16] L. ILLUSIE , Report on crystalline cohomology , Proc. of Symp. in pure Math. 29, 459-478 ( 1975 ). MR 52 #13845 | Zbl 0326.14005 · Zbl 0326.14005
[17] L. ILLUSIE , Complexe de De Rham-Witt , Astérisque 63, 83-112 ( 1979 ). MR 81j:14012 | Zbl 0446.14008 · Zbl 0446.14008
[18] L. ILLUSIE , Complexe de De Rham-Witt et cohomologie cristalline , Ann. Scient. Ec. Norm. Sup. 12, 501-661 ( 1979 ). Numdam | MR 82d:14013 | Zbl 0436.14007 · Zbl 0436.14007
[19] L. ILLUSIE , M. RAYNAUD , Les suites spectrales associées au complexe de De Rham-Witt , Publ. Math. I.H.E.S. 57, 73-212 ( 1983 ). Numdam | MR 85e:14026 | Zbl 0538.14012 · Zbl 0538.14012 · doi:10.1007/BF02698774
[20] L. ILLUSIE , Finiteness, duality, and Künneth theorems in the cohomology of the De Rham-Witt complex , in ”Algebraic Geometry”, Lecture Notes in Math. 1016, 20-72 ( 1983 ). MR 85m:14033 | Zbl 0538.14013 · Zbl 0538.14013
[21] N. KATZ , Travaux de Dwork , Séminaire Bourbaki 1971-72, exposé 409, Lecture Notes in Math. 317, 167-200, Springer Verlag ( 1973 ). Numdam | MR 58 #16672 | Zbl 0259.14007 · Zbl 0259.14007
[22] R. KIEHL , Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie , Inventiones Math. 2, 256-273 ( 1967 ). MR 35 #1834 | Zbl 0202.20201 · Zbl 0202.20201 · doi:10.1007/BF01425404
[23] S. LUBKIN , Finite generation of lifted p-adic homology with compact supports , Journ. of Number Theory 11, 412-464 ( 1979 ). MR 80h:14008 | Zbl 0445.14011 · Zbl 0445.14011 · doi:10.1016/0022-314X(79)90010-6
[24] P. MONSKY , G. WASHNITZER , Formal cohomology: I , Annals of Math. 88, 181-217 ( 1968 ). MR 40 #1395 | Zbl 0162.52504 · Zbl 0162.52504 · doi:10.2307/1970571
[25] P. MONSKY , Formal cohomology : II , Annals of Math. 88, 218-238 ( 1968 ). Zbl 0162.52601 · Zbl 0162.52601 · doi:10.2307/1970572
[26] P. MONSKY , Formal cohomology : III. Fixed point theorems , Annals of Math. 93, 315-343 ( 1971 ). MR 48 #296 | Zbl 0213.47501 · Zbl 0213.47501 · doi:10.2307/1970777
[27] A. OGUS , F-isocrystals and De Rham cohomology II , Duke Math. Journal 51, n^\circ 4, p. 765-850 ( 1984 ). Article | MR 86j:14012 | Zbl 0584.14008 · Zbl 0584.14008 · doi:10.1215/S0012-7094-84-05136-6
[28] M. RAYNAUD , Géométrie analytique rigide , Bull. Soc. Math. France, Mémoire 39, 40, 319-327 ( 1974 ). Numdam | MR 57 #10012 | Zbl 0299.14003 · Zbl 0299.14003
[29] P. ROBBA , Index of p-adic differential operators: III. Application to twisted exponential sums , Astérisque 119-120, p. 191-266 ( 1984 ). MR 86e:12011 | Zbl 0548.12015 · Zbl 0548.12015
[30] S. SPERBER , p-adic hypergeometric functions and their cohomology , Duke Math. Journ. 44, 535-589 ( 1977 ). Article | MR 57 #16307 | Zbl 0408.12025 · Zbl 0408.12025 · doi:10.1215/S0012-7094-77-04424-6
[31] S. SPERBER , Congruence properties of the hyperkloosterman sum , Compositio Mathematica 40, 3-33 ( 1980 ). Numdam | MR 81j:10059 | Zbl 0444.12014 · Zbl 0444.12014
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