One of the most important mathematical achievements of the past several
decades has been A. Grothendieck's work on algebraic geometry. In the
early 1960s, he and M. Artin introduced étale cohomology in order to
extend the methods of sheaf-theoretic cohomology from complex varieties
to more general schemes. This work found many applications, not only in
algebraic geometry, but also in several different branches of number
theory and in the representation theory of finite and p-adic groups.
Yet until now, the work has been available only in the original massive
and difficult papers. In order to provide an accessible introduction to
étale cohomology, J. S. Milne offers this more elementary account
covering the essential features of the theory.
The author begins with a review of the basic properties of flat and
étale morphisms and of the algebraic fundamental group. The next two
chapters concern the basic theory of étale sheaves and elementary étale
cohomology, and are followed by an application of the cohomology to the
study of the Brauer group. After a detailed analysis of the cohomology
of curves and surfaces, Professor Milne proves the fundamental theorems
in étale cohomology -- those of base change, purity, Poincaré duality,
and the Lefschetz trace formula. He then applies these theorems to show
the rationality of some very general L-series.
Originally published in 1980.
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