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 tale 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 tale 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 tale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of tale sheaves and elementary tale 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 tale cohomology -- those of base change, purity, Poincar 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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