Abstract
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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
One of them is missing in [2]: the coherence of the sheaf \(\mathcal{O}_{Z_{\mathrm{G}}}\) itself. It is proven in [4, Lemma 0.1]. Note that it was pointed out to the author by Jérôme Poineau that there is a mistake in this proof: it establishes that a surjection \(\mathcal{O}^{n} \rightarrow \mathcal{O}\) has a locally finitely generated kernel, though in order to get the coherence, this finiteness claim should be established for any, i.e. non necessarily surjective, map \(\mathcal{O}^{n} \rightarrow \mathcal{O}\); but it turns out that the proof doesn’t make any use of this inaccurate surjectivity assumption.
References
V. Berkovich, Spectral Theory and Analytic Geometry Over Non-archimedean Fields. Mathematical Surveys and Monographs, vol. 33 (AMS, Providence, RI, 1990)
V. Berkovich, Étale cohomology for non-archimedean analytic spaces. Inst. Hautes Études Sci. Publ. Math. 78, 5–161 (1993)
V. Berkovich, Vanishing cycles for formal schemes. Invent. Math. 115(3), 539–571 (1994)
A. Ducros, Les espaces de Berkovich sont excellents. Ann. Inst. Fourier 59(4), 1407–1516 (2009)
R. Kiehl, Der Endlichkeitsatz für eingentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math. 2, 191–214 (1967)
M. Temkin, On local properties of non-Archimedean analytic spaces. Math. Annalen 318, 585–607 (2000)
M. Temkin, On local properties of non-Archimedean analytic spaces. II. Isr. J. Math. 140, 1–27 (2004)
A. Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscripta Math. 123(4), 381–451 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Ducros, A. (2015). Cohomological Finiteness of Proper Morphisms in Algebraic Geometry: A Purely Transcendental Proof, Without Projective Tools. In: Ducros, A., Favre, C., Nicaise, J. (eds) Berkovich Spaces and Applications. Lecture Notes in Mathematics, vol 2119. Springer, Cham. https://doi.org/10.1007/978-3-319-11029-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-11029-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11028-8
Online ISBN: 978-3-319-11029-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)