Nonabelian basechange theorems and étale homotopy theory

• Verfasst von: Haine, Peter [VerfasserIn]
• Verfasst von: Holzschuh, Tim [VerfasserIn]
• Verfasst von: Wolf, Sebastian [VerfasserIn]
• Titel: Nonabelian basechange theorems and étale homotopy theory
• Verf.angabe: Peter J. Haine, Tim Holzschuh, Sebastian Wolf
• E-Jahr: 2024
• Jahr: December 2024
• Umfang: 45 S.
• Fussnoten: Gesehen am 16.06.2025
• Titel Quelle: Enthalten in: Journal of topology
• Ort Quelle: Hoboken, NJ : Wiley, 2008
• Jahr Quelle: 2024
• Band/Heft Quelle: 17(2024), 4 vom: Dez., Artikel-ID e70009, Seite 1-45
• ISSN Quelle: 1753-8424
• DOI: doi:10.1112/topo.70009
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1928261841

Abstract

This paper has two main goals. First, we prove nonabelian refinements of basechange theorems in étale cohomology (i.e., prove analogues of the classical statements for sheaves of spaces). Second, we apply these theorems to prove a number of results about the étale homotopy type. Specifically, we prove nonabelian refinements of the smooth basechange theorem, Huber-Gabber affine analogue of the proper basechange theorem, and Fujiwara-Gabber rigidity theorem. Our methods also recover Chough's nonabelian refinement of the proper basechange theorem. Transporting an argument of Bhatt-Mathew to the nonabelian setting, we apply nonabelian proper basechange to show that the profinite étale homotopy type satisfies arc-descent. Using nonabelian smooth and proper basechange and descent, we give rather soft proofs of a number of Künneth formulas for the étale homotopy type.