Non-Archimedean Analysis
A Systematic Approach to Rigid Analytic Geometry
(Sprache: Englisch)
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:So eine Illrbeit witb eigentIid) nie rertig, man muli iie fur fertig erfHiren, wenn man nad) 8eit nnb Umftiinben bas moglid)fte get an qat. (@oetqe,
Inhaltsverzeichnis zu „Non-Archimedean Analysis “
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.
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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. Existenceof 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
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Bibliographische Angaben
- Autoren: S. Bosch , U. Güntzer , R. Remmert
- 1984, 1984, 436 Seiten, Maße: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 3540125469
- ISBN-13: 9783540125464
- Erscheinungsdatum: 01.05.1984
Sprache:
Englisch
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