# Etale cohomology and the Weil conjecture by E. Freitag, Rinhardt Kiehl

By E. Freitag, Rinhardt Kiehl

This ebook is anxious with probably the most vital advancements in algebraic geometry over the past a long time. In 1949 Andr? Weil formulated his recognized conjectures concerning the numbers of strategies of diophantine equations in finite fields. He himself proved his conjectures via an algebraic thought of Abelian types within the one-variable case. In 1960 seemed the 1st bankruptcy of the "El?ments de G?ometrie Alg?braique" par A. Grothendieck (en collaboration avec J. Dieudonn?). In those "El?ments" Grothendieck advanced a brand new beginning of algebraic geometry with the declared target to come back to an evidence of the Weil conjectures via a brand new algebraic cohomology idea. Deligne succeded in proving the Weil conjectures at the foundation of Grothendiecks rules. the purpose of this "Ergebnisbericht" is to increase as self-contained as attainable and as brief as attainable Grothendiecks 1-adic cohomology conception together with Delignes monodromy idea and to provide his unique facts of the Weil conjectures.

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He showed that the pair (m, k) classifies the orientable (compact) surfaces uniquely (Jordan, 1866, p. 85). For the proof he used dissection of the surfaces along the recurrent cuts and additional cross cuts and topological maps of the resulting simply connected pieces. Jordan, in contrast to Mobius, was aware of the connections between the topological theory of surfaces and complex function theory. Another aspect of his work on surfaces, the study of homotopy classes of closed paths, very likely was motivated by this context, although he did not remark so expHcidy and left it to the reader to realize it.

J. Brouwer, Collected Works, Vol. 2, H. , Amsterdam (1976). 24 T. Crilly with D. Johnson [3] G. Cantor, Georg Canton Briefe, H. Meschkowski and W. Nilson, eds, Berlin (1991). -L. Chabert, Un demi-siecle defractales: 1870-1920, Historia Mathematica 17 (1990), 339-365. W. Dauben, The invariance of dimension', problems in the early development of set theory and topology, Historia Mathematica 2 (1975), 273-288. W. Dauben, Georg Cantor, his Mathematics and Philosophy of the hifinite, Princeton (1979).

In slightly modernized reading Riemann thus worked with a geometrical description of bordance homology of submanifolds in F modulo 2, or, in another translation, with simplicial homology, if F is simplicially decomposed by cuts along the curves c/ such that the latter represent 2-cycles of the decomposition. Indeed Riemann showed that there is a welldetermined number n of homologically independent curves, independent of the choice of the specific realization of the curve system, and that in the case of his surfaces this number is even, n = 2p (Riemann's notation (Riemann, 1857, p.