Grundlehren Der Mathematischen Wissenschaften

A. Linear Ultrametric Analysis and Valuation Theory.- 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 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.- 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 An.- 2.2.2. The modules A(I)A(?)c(A) and b(A).- 2.2.3. Structure of L(cI(A), M).- 2.2.4. The ring A [Y1, Y2, ...] 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.- 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 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.- 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.- B. Affinoid algebras.- 5. Strictly convergent power series.- 5.1. Definition and elementary properties of Tn and T?n.- 5.1.1. Description of Tn.- 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. Tn as a function algebra.- 5.2. Weierstrass-Ruckert theory for Tn.- 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 Tn.- 5.2.7. Finite Tn-modules.- 5.3. Stability of Q(Tn).- 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|.- 6. Affinoid algebras and Finiteness Theorems.- 6.1. Elementary properties of affinoid algebras.- 6.1.1. The category 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.- C. Rigid analytic geometry.- 7. Local theory of affinoid varieties.- 7.1. Affinoid varieties.- 7.1.1. Max Tn and the unit ball Bn(ka).- 7.1.2. Affinoid sets. Hilbert's Nullstellensatz.- 7.1.3. Closed subspaces of Max Tn.- 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.- 8. ?ech 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. ?ech 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.- 9. Rigid analytic varieties.- 9.1. Grothendieck topologies.- 9.1.1. 6r-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. -modules.- 9.4.2. Associated modules.- 9.4.3. It-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.- Glossary of Notations.