2012/02/29 by Jérôme Poineau · 24 citations
Mathematics · #Advanced Topology and Set Theory #Affine transformation #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number field #Discrete mathematics #Local ring #Mathematics #Noetherian #Pure mathematics #Ring (chemistry) #Ring of integers #Rings, Modules, and Algebras #Sheaf #math.AG #math.NT #msc:13E05 #msc:13H05 #msc:14G22 #msc:14G25 #msc:32B05 #msc:32P05
paper · pdf · doi:10.1007/s00222-013-0451-6
published in Inventiones mathematicae 194(3), 535-590 (Springer Science+Business Media) · v3: Corrected a few mistakes. Corrected the proof of the Weierstrass division theorem 7.3 in the case where the base field is imperfect and trivially valued
arxiv created 2012/10/15 · openalex publication_date 2013/01/24 · arxiv updated 2015/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We investigate the local properties of Berkovich spaces over Z. Using Weierstrass theorems, we prove that the local rings of those spaces are noetherian, regular in the case of affine spaces and excellent. We also show that the structure sheaf is coherent. Our methods work over other base rings (valued fields, discrete valuation rings, rings of integers of number fields, etc.) and provide a unified treatment of complex and p-adic spaces.