Kato-Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes
| Autor: | Sarah Scherotzke, Mattia Talpo, David Carchedi, Nicolò Sibilla |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
math.AT 14F35 profinite spaces CATEGORIES Geological & Geomatics Engineering 01 natural sciences Mathematics::Algebraic Topology 0101 Pure Mathematics 14F35 55P60 55U35 math.AG Mathematics - Algebraic Geometry Mathematics::Group Theory Stack (abstract data type) Mathematics::K-Theory and Homology 0103 physical sciences FOS: Mathematics Algebraic Topology (math.AT) étale homotopy type Category Theory (math.CT) Mathematics - Algebraic Topology 0101 mathematics Equivalence (formal languages) Kato–Nakayama space math.CT QA 55U35 Algebraic Geometry (math.AG) topological stack Mathematics Science & Technology root stack Homotopy 010102 general mathematics TOPOLOGICAL STACKS Mathematics - Category Theory log scheme infinity category 55P60 Physical Sciences 010307 mathematical physics Geometry and Topology Settore MAT/03 - Geometria |
| Zdroj: | Geom. Topol. 21, no. 5 (2017), 3093-3158 |
| ISSN: | 1364-0380 |
| Popis: | For a log scheme locally of finite type over $\mathbb{C}$, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over $\mathbb{C}$, another natural candidate is the profinite \'etale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over $\mathbb{C}$, these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite \'etale homotopy type of its infinite root stack. Comment: Final version. 57 pages |
| Databáze: | OpenAIRE |
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