Étale Cohomology (PMS-33), Volume 33 (Princeton Mathematical Series) by James S. Milne (PDF)

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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.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐When I made a first pass at this book, I was a little annoyed, basically because I found a lot of the opening sections too brief. Since I wanted to learn this etale business, I then set about using the “classical” references…which are in French. At the end of a long year (reading French books), I returned to Milne’s book and now really see that what this book accomplishes is pretty amazing. In a book of only a few hundred pages he has managed to compress many years of difficult mathematics. Most importantly, he was able to convey the geometric and arithmetic ideas behind the results in fairly down to earth terms. He also includes some nice stuff on torsors and the Brauer group. I would recommend using this book. If you want to know more about something, you can always find other things in SGA 4, 4 1/2 and Freitag & Kiehl’s book.

⭐This book is a rigorous overview of an approach to the study of schemes that uses a generalization of the complex topology called the etale topology. For the case of of ordinary complex algebraic varieties, the etale topology is simply the complex topology, which gives more information about the variety than the Zariski topology. The etale topology gives rise to sheaf and cohomology theories for schemes that parallels that of the cohomology of complex manifolds. These concepts are discussed in great detail in this book, and the reader is expected to have a substantial background in algebra, algebraic geometry, and topology. With this background, the reader will encounter an exceptionally well-written book full of fascinating ideas and constructions. There is also a set of notes on etale cohomology written by the author that he has posted on the Web. The notes compliment this book and treat the case of varieties rather than schemes. When one is dealing with a complex algebraic variety, most of the usual concepts from algebraic topology carry over. For example, the dimensions of the rational cohomology groups of the variety, namely the Betti numbers, can be defined. For a variety over a general algebraically closed field, one has to deal with the Zariski topology, which is not fine enough for the usual techniques of algebraic topology. One example of this are the integer cohomology groups, which for an irreducible variety are all zero except in dimension zero. When one uses the etale topology though, the corresponding cohomology groups give the correct Betti numbers and the standard results of algebraic topology carry over. That the etale topology is a generalization of the complex topology is easier to see in the context of varieties rather than schemes as is done in this book. An etale covering of a nonsingular complex algebraic variety can be refined by a covering for the complex topology. The reader familiar with Riemann surfaces should keep in mind the concept of an unramified covering when studying this chapter. The first chapter emphasizes the role of etale morphisms and their role in defining the etale topology. To reinforce this connection even further, the author asks the reader to show that a morphism between two smooth varieties over a field is etale if and only if the morphism induces an isomorphism on the tangent spaces. When going over to a general scheme, this geometric picture is lost, and one must rely completely on algebraic constructions. But the author does show effectively that etale morphisms for schemes are essentially local isomorphisms in a sense. He also discusses the fundamental group of a scheme in this chapter. When reading this section, it is best to think of the fundamental group from ordinary topology in terms of the universal covering space instead of simple connectedness as it is the former concept that is employed to define the fundamental group of a scheme. The author turns to sheaf theory in the next chapter and shows how etale topology does give exactness for sequences of sheaves that are not exact in the Zariski topology. The notion of a flat topology, even finer than the etale topology, is integrated into the discussion. The author gives a nice example of a sequence involving a scheme of characteristic p that is exact relative to the flat topology, but not the etale or Zariski topologies. The next chapter moves on to the cohomology of sheaves on flat and etale sites. He convinces the reader right away that the sheaf category is more general than the abelian category by showing that the former does not have enough projectives. Cech cohomology is introduced mostly as a device to compute the cohomology groups, which is difficult to do directly when expressed in terms of derived functors. The author takes greatcare in explaining the connection between these different approaches, for example showing that for a separated scheme over a quasi-coherent sheaf, they coincide. Also, several very insightful examples of the actual calculation of the cohomology groups relative to the etale topology are given. In addition, the author compares flat vs. Zariski, flat vs etale, and etale vs complex cohomologies. The next chapter covers the Brauer group, which was introduced by Grothendieck as a generalization of central simple algebras over fields. The Brauer group is defined in terms of equivalence classes of Azumaya algebras, with the cohomological Brauer group defined as the torsion part of the second etale cohomology group. There is an injection of the Brauer group into the cohomological Brauer group, and this chapter outlines what was known at the time of publication when this is a bijection. It is now known that for schemes of dimension less than or equal to 1, regular surfaces, Abelian varieties, unions of affine schemes with affine intersection, smooth toric varieties, and separated geometrically normal algebraic surfaces that such a bijection exists. Starting with the construction of a Lefschetz pnecil, so powerful in symplectic geometry, the calculation of the cohomology curves and surfaces is taken up in the next chapter. The author’s proofs are very concise but understandable and he explains the need for using constructible sheaves in obtaining the usual Euler characteristic and Poincare duality results from algebraic topology.The situation in characteristic p is treated in detail. This sets the stage for the last chapter which proves the Kunneth formula and the Lefschetz fixed point formula. The all-important L-series is proved to be rational when it arises from representations of Galois groups. This chapter should satisfy algebraic topologists who are curious as to what extent the usual results in their area carry over to schemes.

⭐Milne’s account of ‘etale cohomology is sometimes useful—in that it proves various little lemmata and criteria that come handy in practice—but disorganised the rest of the time. What’s more, the proofs are often just sketches of the sort: “To see this, use the argument of Theorem 3.66 combined with the devissage of SGA 4, exp. XII, Prop. 3.14 to reduce to the case of curves; now the claim follows from my comments on page 45.” (The previous sentence is not a direct quotation from the book, so it might suffer from mild exaggeration on my part.) Furthermore, the book doesn’t cover Deligne’s proof of the Weil conjectures.If trying to learn the theory for the first time, I’d recommend you read Deligne’s extremely lucid and concise SGA 4 1/2. As a good exercise for your new-found knowledge, then, try to go through the same author’s “Weil-II” paper. This should probably serve you better than reading Milne’s Etale Cohomology.

⭐Nice book.

⭐以前，ハードカバーの物を購入して所持していたのですが行方不明になっていたので買い直しました.お手頃価格になっていると思います.

⭐

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