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).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
