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Among the many differences between classical and p-adic objects, those related to differential equations occupy a special place. For example, a closed p-adic analytic one-form defined on a simply-connected domain does not necessarily have a primitive in the class of analytic functions. In the early 1980s, Robert Coleman discovered a way to construct primitives of analytic one-forms on certain smooth p-adic analytic curves in a bigger class of functions. Since then, there have been several attempts to generalize his ideas to smooth p-adic analytic spaces of higher dimension, but the spaces considered were invariably associated with algebraic varieties. This book aims to show that every smooth p-adic analytic space is provided with a sheaf of functions that includes all analytic ones and satisfies a uniqueness property. It also contains local primitives of all closed one-forms with coefficients in the sheaf that, in the case considered by Coleman, coincide with those he constructed. In consequence, one constructs a parallel transport of local solutions of a unipotent differential equation and an integral of a closed one-form along a path so that both depend nontrivially on the homotopy class of the path. Both the author's previous results on geometric properties of smooth p-adic analytic spaces and the theory of isocrystals are further developed in this book, which is aimed at graduate students and mathematicians working in the areas of non-Archimedean analytic geometry, number theory, and algebraic geometry.

p-adic analysis. --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Abelian category. --- Acting in. --- Addition. --- Aisle. --- Algebraic closure. --- Algebraic curve. --- Algebraic structure. --- Algebraic variety. --- Allegory (category theory). --- Analytic function. --- Analytic geometry. --- Analytic space. --- Archimedean property. --- Arithmetic. --- Banach algebra. --- Bertolt Brecht. --- Buttress. --- Centrality. --- Clerestory. --- Commutative diagram. --- Commutative property. --- Complex analysis. --- Contradiction. --- Corollary. --- Cosmetics. --- De Rham cohomology. --- Determinant. --- Diameter. --- Differential form. --- Dimension (vector space). --- Divisor. --- Elaboration. --- Embellishment. --- Equanimity. --- Equivalence class (music). --- Existential quantification. --- Facet (geometry). --- Femininity. --- Finite morphism. --- Formal scheme. --- Fred Astaire. --- Functor. --- Gavel. --- Generic point. --- Geometry. --- Gothic architecture. --- Homomorphism. --- Hypothesis. --- Imagery. --- Injective function. --- Irreducible component. --- Iterated integral. --- Linear combination. --- Logarithm. --- Marni Nixon. --- Masculinity. --- Mathematical induction. --- Mathematics. --- Mestizo. --- Metaphor. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Neuroticism. --- Noetherian. --- Notation. --- One-form. --- Open set. --- P-adic Hodge theory. --- P-adic number. --- Parallel transport. --- Patrick Swayze. --- Phrenology. --- Politics. --- Polynomial. --- Prediction. --- Proportion (architecture). --- Pullback. --- Purely inseparable extension. --- Reims. --- Requirement. --- Residue field. --- Rhomboid. --- Roland Barthes. --- Satire. --- Self-sufficiency. --- Separable extension. --- Sheaf (mathematics). --- Shuffle algebra. --- Subgroup. --- Suggestion. --- Technology. --- Tensor product. --- Theorem. --- Transept. --- Triforium. --- Tubular neighborhood. --- Underpinning. --- Writing. --- Zariski topology.

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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.

Ordered algebraic structures --- 512.73 --- 512.66 --- Geometry, Algebraic --- Homology theory --- Sheaf theory --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Cohomology theory --- Contrahomology theory --- Algebraic geometry --- Geometry --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- Geometry, Algebraic. --- Homology theory. --- Sheaf theory. --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Abelian category. --- Abelian group. --- Adjoint functors. --- Affine variety. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic cycle. --- Algebraic equation. --- Algebraic space. --- Algebraically closed field. --- Artinian. --- Automorphism. --- Base change. --- Brauer group. --- CW complex. --- Cardinal number. --- Category of sets. --- Central simple algebra. --- Chow's lemma. --- Closed immersion. --- Codimension. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Complex number. --- Dedekind domain. --- Derived category. --- Diagram (category theory). --- Direct limit. --- Discrete valuation ring. --- Divisor. --- Epimorphism. --- Equivalence class. --- Existential quantification. --- Fibration. --- Field of fractions. --- Fine topology (potential theory). --- Finite field. --- Finite morphism. --- Flat morphism. --- Functor. --- Fundamental class. --- Fundamental group. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Generic point. --- Group scheme. --- Gysin sequence. --- Henselian ring. --- Identity element. --- Inclusion map. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Invertible sheaf. --- Isomorphism class. --- Lefschetz pencil. --- Local ring. --- Maximal ideal. --- Module (mathematics). --- Morphism of schemes. --- Morphism. --- Noetherian. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime ideal. --- Prime number. --- Principal homogeneous space. --- Profinite group. --- Projection (mathematics). --- Projective variety. --- Quasi-compact morphism. --- Residue field. --- Riemann surface. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Stein factorization. --- Subalgebra. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Surjective function. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Torsion sheaf. --- Torsor (algebraic geometry). --- Vector bundle. --- Weil conjecture. --- Yoneda lemma. --- Zariski topology. --- Zariski's main theorem. --- Geometrie algebrique --- Cohomologie