脡tale Cohomology
A. Grothendieck鈥檚 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 茅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 茅tale cohomology, covering the essential features of the theory.
Milne begins with a review of the basic properties of flat and 茅tale morphisms and the algebraic fundamental group. He then turns to the basic theory of 茅tale sheaves and elementary 茅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 茅tale cohomology鈥攖hose of base change, purity, Poincar茅 duality, and the Lefschetz trace formula鈥攁nd applies these theorems to show the rationality of some very general L-series.