Cohomological Finiteness of Proper Morphisms in Algebraic Geometry: A Purely Transcendental Proof, Without Projective Tools

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• 作者：Antoine Ducros (16)
  
• 刊名：Lecture Notes in Mathematics
• 出版年：2015
• 出版时间：2015
• 年：2015
• 卷：2119
• 期：1
• 页码：135-140
• 全文大小：137 KB
• 参考文献：1. V. Berkovich, / Spectral Theory and Analytic Geometry Over Non-archimedean Fields. Mathematical Surveys and Monographs, vol. 33 (AMS, Providence, RI, 1990)
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  4. A. Ducros, Les espaces de Berkovich sont excellents. Ann. Inst. Fourier 59(4), 1407-516 (2009) CrossRef
  5. R. Kiehl, Der Endlichkeitsatz für eingentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math. 2, 191-14 (1967) CrossRef
  6. M. Temkin, On local properties of non-Archimedean analytic spaces. Math. Annalen 318, 585-07 (2000) CrossRef
  7. M. Temkin, On local properties of non-Archimedean analytic spaces. II. Isr. J. Math. 140, 1-7 (2004) CrossRef
  8. A. Thuillier, Géométrie toro?dale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscripta Math. 123(4), 381-51 (2007)
  
• 作者单位：Antoine Ducros (16)
  
  16. Institut de Mathématiques de Jussieu, Université Paris 6, 4 place Jussieu, 75252, Paris Cedex 05, France
  
• ISSN：1617-9692

文摘

In this short note, we explain how one can prove without projective tools that the higher direct images of a coherent sheaf under a map between two schemes of finite type over a field are coherent. The proof consists in endowing the ground field with the trivial absolute value and using the corresponding finiteness theorem for Berkovich spaces (after having proven at hand a suitable GAGA-principle). The latter theorem comes itself from a theorem of Kiehl in rigid geometry, whose proof is based upon the theory of completely continuous maps between p-adic Banach spaces (in the spirit of Cartan and Serre’s proof of the finiteness of coherent cohomology on a compact complex analytic space).