Leseprobe
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Étale Cohomology
James S. Milne
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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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Integral domain, Finite field, Cohomology ring, Codimension, Galois extension, Complex number, Galois cohomology, Abelian category, Subgroup, Zariski's main theorem, Isomorphism class, Yoneda lemma, Surjective function, Cokernel, Fibration, G-module, Algebraic cycle, Algebraically closed field, Theorem, Subalgebra, Algebraic equation, Residue field, Morphism, Weil conjecture, Projection (mathematics), Functor, Group scheme, Sheaf of modules, Subset, Projective variety, Sheaf (mathematics), Subring, Alexander Grothendieck, Brauer group, Existential quantification, Local ring, Open set, Vector bundle, Affine variety, Galois group, Intersection (set theory), Principal homogeneous space, Algebraic closure, Torsor (algebraic geometry), Field of fractions, Presheaf (category theory), Lefschetz pencil, Diagram (category theory), Profinite group, Subcategory, Torsion sheaf, Stein factorization, Henselian ring, Closed immersion, Cohomology, Dedekind domain, Noetherian, Chow's lemma, Category of sets, Spectral sequence, Finite morphism, Commutative diagram, Fundamental group, Topological space, Topology, Algebraic space, Direct limit, Zariski topology, Base change, Invertible sheaf