Nonabelian basechange theorems & étale homotopy theory
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909217761263616 |
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| author | Haine, Peter J. Holzschuh, Tim Wolf, Sebastian |
| author_facet | Haine, Peter J. Holzschuh, Tim Wolf, Sebastian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_00938 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonabelian basechange theorems & étale homotopy theory Haine, Peter J. Holzschuh, Tim Wolf, Sebastian Algebraic Geometry Algebraic Topology 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. |
| title | Nonabelian basechange theorems & étale homotopy theory |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2304.00938 |