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2017 Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes
David Carchedi, Sarah Scherotzke, Nicolò Sibilla, Mattia Talpo
Geom. Topol. 21(5): 3093-3158 (2017). DOI: 10.2140/gt.2017.21.3093

Abstract

For a log scheme locally of finite type over , 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 , another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over , 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 étale homotopy type of its infinite root stack.

Citation

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David Carchedi. Sarah Scherotzke. Nicolò Sibilla. Mattia Talpo. "Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes." Geom. Topol. 21 (5) 3093 - 3158, 2017. https://doi.org/10.2140/gt.2017.21.3093

Information

Received: 25 April 2016; Revised: 28 September 2016; Accepted: 11 November 2016; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06774941
MathSciNet: MR3687115
Digital Object Identifier: 10.2140/gt.2017.21.3093

Subjects:
Primary: 14F35, 55U35
Secondary: 55P60

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.21 • No. 5 • 2017
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