Spectral Theory and Analytic Geometry Over Non-Archimedean FieldsThe purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and p -adic analysis. |
Contents
Introduction | 1 |
The Spectrum of a Commutative Banach Ring | 11 |
Affinoid Spaces | 21 |
Analytic Spaces | 47 |
Analytic Curves | 75 |
Analytic Groups and Buildings | 93 |
The Homotopy Type of Certain Analytic Spaces | 115 |
Spectral Theory | 127 |
Perturbation Theory | 139 |
The Dimension of a Banach Algebra | 153 |
References | 161 |
Other editions - View all
Spectral Theory and Analytic Geometry over Non-Archimedean Fields Vladimir G. Berkovich Limited preview - 2012 |
Spectral Theory and Analytic Geometry Over Non-Archimedean Fields Vladimir G. Berkovich No preview available - 1990 |
Common terms and phrases
abelian variety affinoid neighborhood affinoid subdomain assume Banach ring Banach space bounded homomorphism bounded linear operators canonical map closed disc closed immersion closed subset coherent coincides commutative Banach k-algebra COROLLARY corresponding countable defined denote dim(X dimension element ƒ embedding endpoints epimorphism equal equivalent everywhere dense f₁ fibre finite morphisms functor G-topology g₁ Gal(K/k group G Hence Hom(Î homomorphism homotopy Int(Y/X inverse image invertible irreducible isomorphism k-algebra with identity k-analytic group k-analytic space Lemma Let f locally ringed spaces maximal module multiplicative seminorm non-Archimedean field nonempty nontrivial norm normal open immersion open neighborhood open subset phism polynomial PROOF proper Proposition r₁ reduction resp seminorm sheaf simply connected spectrum strictly k-affinoid algebra subalgebra subgroup of G subspace suffices to show Suppose surjective T₁ Theorem topology torus trivial valuation field Weierstrass XÔK zero