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The cohomology of Monsky and Washnitzer. (English) Zbl 0606.14018
The paper gives a very nice introduction to Monsky-Washnitzer cohomology. It begins by the definition of MW cohomology and some of its basic properties, such as construction of the Frobenius map and the Lefschetz fixed point formula for it. The exposition has been simplified and somewhat extended with the aid of the Artin approximation theorem and some rigid analysis. The article then goes on to describe some more specific topics and ends by discussing Dwork’s work on the cohomology of the family of elliptic curves, its unit root part etc. This article can be recommended as a short introduction to anyone interested in these subjects.
Reviewer: T.Ekedahl

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14G20 Local ground fields in algebraic geometry 14B12 Local deformation theory, Artin approximation, etc. 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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##### References:
 [1] A. Adolphson - An index theorem for p-adic differential operators . Trans. Amer. Math. Soc. 216 ( 1976 ) 297-293. MR 52 #8127 | Zbl 0297.47039 · Zbl 0297.47039 · doi:10.2307/1997699 [2] M. Artin - On the solutions of analytic equations . Invent. math. 5, 277-291 ( 1968 ). MR 38 #344 | Zbl 0172.05301 · Zbl 0172.05301 · doi:10.1007/BF01389777 · eudml:141922 [3] R. Berger - Über Verschiedene Differentenbegriffe , Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Jahrg. 1960 . Abh. 1. Zbl 0111.24603 · Zbl 0111.24603 [4] S. Bosch - A rigid analytic version of M. Artin’s theorem on analytic equations . Math.Ann. 255, 395-404 ( 1981 ). MR 82k:32067 | Zbl 0462.14002 · Zbl 0462.14002 · doi:10.1007/BF01450712 · eudml:182848 [5] C.H. Clemers - A Scrapbook of Complex Curve Theory . Plenum Press. 1980 . Zbl 0456.14016 · Zbl 0456.14016 [6] B. Dwork - A deformation theory for the zeta function of a hypersurface . Proc.Int.Cong.Math. ( 1962 ) 247-259. Zbl 0196.53302 · Zbl 0196.53302 [7] B. Dwork - Lectures on p-adic Differential Equations . Grundlehren. N^\circ 253, Springer Verlag 1982 . MR 84g:12031 | Zbl 0502.12021 · Zbl 0502.12021 [8] N. Katz - On the differential equations satisfied by period matrices . Publ. I.H.E.S. N^\circ 35 - 1968 . Numdam | MR 39 #4168 | Zbl 0159.22502 · Zbl 0159.22502 · doi:10.1007/BF02698924 · numdam:PMIHES_1968__35__71_0 · eudml:103887 [9] N. Katz - Travaux de Dwork - Séminaire Bourbaki . 1971 / 1972 n^\circ 409. Numdam | Zbl 0259.14007 · Zbl 0259.14007 · numdam:SB_1971-1972__14__167_0 · eudml:109810 [10] N. Katz - Algebraic Solutions of Differential Equations . (p-Curvature and the Hodge Filtration/Invent.math. 18, 1-118 ( 1972 ). MR 49 #2728 | Zbl 0278.14004 · Zbl 0278.14004 · doi:10.1007/BF01389714 · eudml:142177 [11] P. Monsky , G. Washnitzer - Formal cohomology I . Annals of Math. 1968 . MR 40 #1395 | Zbl 0162.52504 · Zbl 0162.52504 · doi:10.2307/1970571 [12] P. Monsky - Formal cohomology II and III . Annals of Math. 1968 and 1971 . Zbl 0162.52601 · Zbl 0162.52601 · doi:10.2307/1970572 [13] P. Monsky - p-adic analysis and zeta-functions 1970 . Lectures at Kyoto-University. MR 44 #215 | Zbl 0256.14009 · Zbl 0256.14009 [14] D. Reich - A p-adic fixed point formula , Amer.J.Math. 91 ( 1969 ) 835-850. MR 41 #210 | Zbl 0213.47502 · Zbl 0213.47502 · doi:10.2307/2373354 [15] R. Elkik - Solutions d’équations à coefficients dans un anneau hensélien . Ann.Scient.Ec.Norm.Syp. 6, n^\circ 4, 553-604 ( 1973 ). Numdam | MR 49 #10692 | Zbl 0327.14001 · Zbl 0327.14001 · numdam:ASENS_1973_4_6_4_553_0 · eudml:81927 [16] R. Hartshorne - Algebraic Geometry . GTM 52, Springer Verlag 1977 . MR 57 #3116 | Zbl 0367.14001 · Zbl 0367.14001
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