Haine, Peter J. and Holzschuh, Tim and Wolf, Sebastian (2024) The Fundamental Fiber Sequence in etale Homotopy Theory. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2024 (1). pp. 175-196. ISSN 1073-7928, 1687-0247
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Let k be a field with separable closure k & macr; superset of k, and let X be a qcqs k-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences Pi(et)(<infinity)(X-k & macr;) -> Pi(et )(<infinity)(X) -> BGal(k & macr;/k) and Pi(et)(<infinity)(X-k & macr;) -> Pi(et)(<infinity)(X) -> BGal(k & macr;/k) of protruncated and profinite etale homotopy types are fiber sequences. This gives a common conceptual reason for the following two phenomena: first, the higher etale homotopy groups of X and the geometric fiber X-k & macr; are isomorphic, and second, if X-k & macr; is connected, then the sequence of profinite etale fundamental groups 1 -> pi circumflex expressionccent (et)(1) (X-k & macr;) -> pi circumflex expressionccent (et)(1)(X) -> Gal(k & macr;/k) -> 1 is exact. It also proves the analogous results for the groupe fondamental & eacute;largi of SGA3.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | MODEL STRUCTURE |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 20 Mar 2024 07:01 |
| Last Modified: | 04 Mar 2025 09:38 |
| URI: | https://pred.uni-regensburg.de/id/eprint/60015 |
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