On Serre-duality for coherent sheaves on rigid-analytic spaces
type: Item
instance of: scholarly article
title: On Serre-duality for coherent sheaves on rigid-analytic spaces
zbMATH Open document ID: 0880.32005
DOI: 10.1007/BF02677468
author: Peter Beyer
published in: Manuscripta Mathematica
publication date: 1998-01-08T00:00:00Z
full work available at URL: https://eudml.org/doc/156296
review text: The main purpose of this article is to prove a Serre-duality theorem for coherent sheaves in the rigid-analytic context. Let \(X\) be a smooth proper rigid-analytic variety over \(K\) of dimension \(n\). Let \(\omega_X=\wedge^n\varOmega_{X/K}^1\) be the sheaf of holomorphic \(n-\)forms on \(X\). Then there exists a trace map \(t:H^n(X, \omega_X)\to K\) with the following property: If \(\mathcal F\) is a coherent sheaf on \(X,\) then the morphism \(t\) induces by composition with the canonical map a perfect pairing \[ \text{Hom}(\mathcal F,\omega_X)\times H^n(X,\mathcal F)\to H^n(X,\omega_X)\to K \] of \(K-\)vector spaces. Especially there is a canonical isomorphism \[ H^n(X,\mathcal F)^\lor\widetilde{\longrightarrow}\text{Hom}(\mathcal F, \omega_X). \] The proof is given by using results from algebraic local cohomology.
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 32P05
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 1038308
zbMATH Keywords: analytic sheaves and cohomology groups
zbMATH Keywords: vector bundles
zbMATH Keywords: special sheaves
MaRDI profile type: Publication
cites work: Q3325851
cites work: Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume
cites work: Q3972364
cites work: Q4143433
cites work: Theorem A und B in der nichtarchimedischen Funktionentheorie
cites work: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie
cites work: Q5680385
cites work: Vektorraumbündel über nichtarchimedischen holomorphen Räumen
cites work: Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne
cites work: Q5187397
cites work: Serre duality for rigid analytic spaces
OpenAlex ID: W1994443317
description: scientific article; zbMATH DE number 1038308