Non-archimedean analysis : a systematic approach to rigid analytic geometry

S. Bosch, U. Güntzer, R. Remmert

[目次]

  • .- 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.

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この本の情報

書名 Non-archimedean analysis : a systematic approach to rigid analytic geometry
著作者等 Bosch, Siegfried
Güntzer, U.
Remmert, Reinhold
Guntzer U.
Remmert R.
Bosch S.
シリーズ名 Die Grundlehren der mathematischen Wissenschaften
出版元 Springer-Verlag
刊行年月 1984
ページ数 xii, 436 p.
大きさ 24 cm
ISBN 3540125469
0387125469
NCID BA03076002
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言語 英語
出版国 ドイツ
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