Notions of Stein spaces in non-archimedean geometry
| Autor: | Jérôme Poineau, Marco Maculan |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Algebra and Number Theory 14G22 32E10 010102 general mathematics MathematicsofComputing_GENERAL Structure (category theory) Boundary (topology) Field (mathematics) Characterization (mathematics) Space (mathematics) 01 natural sciences Cohomology [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] Coherent sheaf Mathematics - Algebraic Geometry 0103 physical sciences Sheaf 010307 mathematical physics Geometry and Topology [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] 0101 mathematics [MATH]Mathematics [math] Mathematics |
| Zdroj: | Journal of Algebraic Geometry |
| Popis: | Let $k$ be a non-archimedean complete valued field and $X$ be a $k$-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: 1) for every complete valued extension $k'$ of $k$, every coherent sheaf on $X \times_{k} k'$ is acyclic; 2) $X$ is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex) and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, $X$ is compact); 3) $X$ admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When $X$ has no boundary the characterization is simpler: in~2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in 3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl's definition of Stein space. v2: major revision to handle also the case of spaces with boundary Comment: 31 pages |
| Databáze: | OpenAIRE |
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