Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry

Introduction
Part A. Linear Ultrametric Analysis and Valuation Theory
Chapter 1. Norms and Valuations
1.1. Semi-normed and normed groups
1.1.1. Ultrametric functions
1.1.2. Filtrations
1.1.3. Semi-normed and normed groups. Ultrametric topology
1.1.4. Distance
1.1.5. Strictly closed subgroups
1.1.6. Quotient groups
1.1.7. Completions
1.1.8. Convergent series
1.1.9. Strict homomorphisms and completions
1.2. Semi-normed and normed rings
1.2.1. Semi-normed and normed rings
1.2.2. Power-multiplicative and multiplicative elements
1.2.3. The category ill and the functor A --> A~
1.2.4. Topologically nilpotent elements and complete normed rings
1.2.5. Power-bounded elements
1.3. Power-multiplicative semi-norms
1.3.1. Definition and elementary properties
1.3.2. Smoothing procedures for semi-norms
1.3.3. Standard examples of norms and semi-norms
1.4. Strictly convergent power series
1.4.1. Definition and structure of A(X)
1.4.2. Structure of A(X)
1.4.3. Bounded homomorphisms of A(X)
1.5. Non-Archimedean valuations
1.5.1. Valued rings
1.5.2. Examples
1.5.3. The Gauss-Lemma
1.5.4. Spectral value of monic polynomials
1.5.5. Formal power series in countably many indeterminates
1.6. Discrete valuation rings
1.6.1. Definition. Elementary properties
1.6.2. The example of F. K. Schmidt
1.7. Bald and discrete B-rings
1.7.1. B-rings
1.7.2. Bald rings
1.8. Quasi-Noetherian B-rings
1.8.1. Definition and characterization
1.8.2. Construction of quasi-Noetherian rings
Chapter 2. Normed modules and normed vector spaces
2.1. Normed and faithfully normed modules
2.1.1. Definition
2.1.2. Submodules and quotient modules
2.1.3. Modules of fractions. Completions
2.1.4. Ramification index
2.1.5. Direct sum. Bounded and restricted direct product
2.1.6. The module ℒ(L, M) of bounded A-linear maps
2.1.7. Complete tensor products
2.1.8. Continuity and boundedness
2.1.9. Density condition
2.1.10. The functor M --> M~. Residue degree
2.2. Examples of normed and faithfully normed A-modules
2.2.1. The module A^n
2.2.2. The modules A^(l), A^(INFINITY), c(A) and b(A)
2.2.3. Structure of ℒ(c_i(A), M)
2.2.4. The ring A [[Y_1, Y 2, ... ]] of formal power series
2.2.5. b-separable modules
2.2.6. The functor M --> T(M)
2.3. Weakly cartesian spaces
2.3.1. Elementary properties of normed spaces
2.3.2. Weakly cartesian spaces
2.3.3. Properties of weakly cartesian spaces
2.3.4. Weakly cartesian spaces and tame modules
2.4. Cartesian spaces
2.4.1. Cartesian spaces of finite dimension
2.4.2. Finite-dimensional cartesian spaces and strictly closed subspaces
2.4.3. Cartesian spaces of arbitrary dimension
2.4.4. Normed vector spaces over a spherically complete field
2.5. Strictly cartesian spaces
2.5.1. Finite-dimensional strictly cartesian spaces
2.5.2. Strictly cartesian spaces of arbitrary dimension
2.6. Weakly cartesian spaces of countable dimension
2.6.1. Weakly cartesian bases
2.6.2. Existence of weakly cartesian bases. Fundamental theorem
2.7. Normed vector spaces of countable type. The Lifting Theorem
2.7.1. Spaces of countable type
2.7.2. Schauder bases. Orthogonality and orthonormality
2.7.3. The Lifting Theorem
2.7.4. Proof of the Lifting Theorem
2.7.5. Applications
2.8. Banach spaces
2.8.1. Definition. Fundamental theorem
2.8.2. Banach spaces of countable type
Chapter 3. Extensions of norms and valuations
3.1. Normed and faithfully normed algebras
3.1.1. A-algebra norms
3.1.2. Spectral values and power-multiplicative norms
3.1.3. Residue degree and ramification index
3.1.4. Dedekind\'s Lemma and a Finiteness Lemma
3.1.5. Power-multiplicative and faithful A-algebra norms
3.2. Algebraic field extensions. Spectral norm and valuations
3.2.1. Spectral norm on algebraic field extensions
3.2.2. Spectral norm on reduced integral K-algebras
3.2.3. Spectral norm and field polynomials
3.2.4. Spectral norm and valuations
3.3. Classical valuation theory
3.3.1. Spectral norm and completions
3.3.2. Construction of inequivalent valuations
3.3.3. Construction of power-multiplicative algebra norms
3.3.4. Hensel\'s Lemma
3.4. Properties of the spectral valuation
3.4.1. Continuity of roots
3.4.2. Krasner\'s Lemma
3.4.3. Example. p-adic numbers
3.5. Weakly stable fields
3.5.1. Weakly cartesian fields
3.5.2. Weakly stable fields
3.5.3. Criterion for weak stability
3.5.4. Weak stability and Japaneseness
3.6. Stable fields
3.6.1. Definition
3.6.2. Criteria for stability
3.7. Banach algebras
3.7.1. Definition and examples
3.7.2. Finiteness and completeness of modules over a Banach algebra
3.7.3. The category M_A
3.7.4. Finite homomorphisms
3.7.5. Continuity of homomorphisms
3.8. Function algebras
3.8.1. The supremum semi-norm on k-algebras
3.8.2. The supremum semi-norm on k-Banach algebras
3.8.3. Banach function algebras
Chapter 4 (Appendix to Part A). Tame modules and Japanese rings
4.1. Tame modules
4.2. A Theorem of Dedekind
4.3. Japanese rings. First criterion for Japaneseness
4.4. Tameness and Japaneseness
Part B. Affinoid algebras
Chapter 5 Strictly convergent power series
5.1. Definition and elementary properties of T_n and T*_n
5.1.1. Description of T_n
5.1.2. The Gauss norm is a valuation and T*_n is a polynomial ring over k
5.1.3. Going up and down between Tn and T*_n
5.1.4. T_n as a function algebra
5.2. Weierstrass-Ruckert theory for T_n
5.2.1. Weierstrass Division Theorem
5.2.2. Weierstrass Preparation Theorem
5.2.3. Weierstrass polynomials and Weierstrass Finiteness Theorem
5.2.4. Generation of distinguished power series
5.2.5. Ruckert\'s theory
5.2.6. Applications of Ruckert\'s theory for T_n
5.2. 7. Finite T_n-modules
5.3. Stability of Q(T_n)
5.3.1. Weak stability
5.3.2. The Stability Theorem. Reductions
5.3.3. Stability of k(X) if |k*| is divisible
5.3.4. Completion of the proof for arbitrary |k*|
Chapter 6. Affinoid algebras and Finiteness Theorems
6.1. Elementary properties of affinoid algebras
6.1.1. The category U of k-affinoid algebras
6.1.2. Noether normalization
6.1.3. Continuity of homomorphisms
6.1.4. Examples. Generalized rings of fractions
6.1.5. Further examples. Convergent power series on general polydiscs
6.2. The spectrum of a k-affinoid algebra and the supremum semi-norm
6.2.1. The supremum semi-norm
6.2.2. Integral homomorphisms
6.2.3. Power-bounded and topologically nilpotent elements
6.2.4. Reduced k-affinoid algebras are Banach function algebras
6.3. The reduction functor A --> A*
6.3.1. Monomorphisms, isometries and epimorphisms
6.3.2. Finiteness of homomorphisms
6.3.3. Applications to group operations
6.3.4. Finiteness of the reduction functor A --> A*
6.3.5. Summary
6.4. The functor A --> A*
6.4.1. Finiteness Theorems
6.4.2. Epimorphisms and isomorphisms
6.4.3. Residue norm and supremum norm. Distinguished k-affinoid algebras and epimorphisms
Part C. Rigid analytic geometry
Chapter 7. Local theory of affinoid varieties
7.1. Affinoid varieties
7.1.1. Max T_n and the unit ball B^n(k_a)
7.1.2. Affinoid sets. Hilbert\'s Nullstellensatz
7.1.3. Closed subspaces of Max T_n
7.1.4. Affinoid maps. The category of affinoid varieties
7.1.5. The reduction functor
7.2. Affinoid subdomains
7.2.1. The canonical topology on Sp A
7.2.2. The universal property defining affinoid subdomains
7.2.3. Examples of open affinoid subdomains
7.2.4. Transitivity properties
7.2.5. The Openness Theorem
7.2.6. Affinoid subdomains and reduction
7.3. Immersions of affinoid varieties
7.3.1. Ideal-adic topologies
7.3.2. Germs of affinoid functions
7.3.3. Locally closed immersions
7.3.4. Runge immersions
7.3.5. Main theorem for locally closed immersions
Chapter 8. Cech cohomology of affinoid varieties
8.1. Cech cohomology with values in a presheaf
8.1.1. Cohomology of complexes
8.1.2. Cohomology of double complexes
8.1.3. Cech cohomology
8.1.4. A Comparison Theorem for Cech cohomology
8.2. Tate\'s Acyclicity Theorem
8.2.1. Statement of the theorem
8.2.2. Affinoid coverings
8.2.3. Proof of the Acyclicity Theorem for Laurent coverings
Chapter 9. Rigid analytic varieties
9.1. Grothendieck topologies
9.1.1. G-topological spaces
9.1.2. Enhancing procedures for G-topologies
9.1.3. Pasting of G-topological spaces
9.1.4. G-topologies on affinoid varieties
9.2. Sheaf theory
9.2.1. Presheaves and sheaves on G-topological spaces
9.2.2. Sheafification of presheaves
9.2.3. Extension of sheaves
9.3. Analytic varieties. Definitions and constructions
9.3.1. Locally G-ringed spaces and analytic varieties
9.3.2. Pasting of analytic varieties
9.3.3. Pasting of analytic maps
9.3.4. Some basic examples
9.3.5. Fibre products
9.3.6. Extension of the ground field
9.4. Coherent modules
9.4.1. O-modules
9.4.2. Associated modules
9.4.3. U-coherent modules
9.4.4. Finite morphisms
9.5. Closed analytic subvarieties
9.5.1. Coherent ideals. The nilradical
9.5.2. Analytic subsets
9.5.3. Closed immersions of analytic varieties
9.6. Separated and proper morphisms
9.6.1. Separated morphisms
9.6.2. Proper morphisms
9.6.3. The Direct Image Theorem and the Theorem on Formal Functions
9.7. An application to elliptic curves
9.7.1. Families of annuli
9.7.2. Affinoid subdomains of the unit disc
9.7.3. Tate\'s elliptic curves
Bibliography
Glossary of Notations
Index