An authoritative introduction to the essential features of tale cohomology

A. Grothendieck's work on algebraic geometry is one of the most important mathematical achievements of the twentieth century. In the early 1960s, he and M. Artin introduced tale cohomology 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. In this classic book, James Milne provides an invaluable introduction to tale cohomology, covering the essential features of the theory.

Milne begins with a review of the basic properties of flat and tale morphisms and the algebraic fundamental group. He then turns to the basic theory of tale sheaves and elementary tale cohomology, 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, Milne proves the fundamental theorems in tale cohomology-those of base change, purity, Poincar duality, and the Lefschetz trace formula-and applies these theorems to show the rationality of some very general L-series.