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Shape Theory in Homotopy Theory and Algebraic Geometry

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posted on 27.11.2018, 00:00 authored by Joseph Berner
This work defines the étale homotopy type in the context of non-archimedean geometry, in both Berkovich’s and Huber’s formalisms. To do this we take the shape of a site’s associated hypercomplete 1-topos. This naturally leads to discussing localizations of the category of pro-spaces. For a prime number p, we introduce a new localization intermediate between profinite spaces and {p}`-profinite spaces. This new category is well suited for comparison theorems when working over a discrete valuation ring of mixed characteristic. We prove a new comparison theorem on the level of topoi for the formalisms of Berkovich and Huber, and prove an analog of smooth-proper base change for nonarchimedean analytic spaces. This provides a necessary result for the non-archimedean analog of Friedlander’s homotopy fiber theorem, which we prove. For a variety over a non-archimedean field, we prove a comparison theorem between the classical étale homotopy type and our étale homotopy type of the variety’s analytification. Finally, we examine certain log formal schemes over the formal spectrum of a complete discrete valuation ring, and compare their Kummer étale homotopy type with the étale homotopy type of the associated non-archimedean analytic space.

History

Advisor

Gillet, Henri

Chair

Gillet, Henri

Department

Mathematics, Statistics, and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Committee Member

Shipley, Brooke Antieau, Ben Lesieutre, John Gepner, David

Submitted date

August 2018

Issue date

15/06/2018

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