Fundamental Exact Sequence for the Pro-Étale Fundamental Group
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866914692511825920 |
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| author | Lara, Marcin |
| author_facet | Lara, Marcin |
| contents | The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual étale fundamental group $π_1^{\mathrm{et}}$ defined in SGA1 and the more general group defined in SGA3. It controls local systems in the pro-étale topology and leads to an interesting class of "geometric covers" of schemes, generalizing finite étale covers.
We prove the homotopy exact sequence over a field for the pro-étale fundamental group of a geometrically connected scheme $X$ of finite type over a field $k$, i.e. that the sequence $$1 \rightarrow π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X) \rightarrow \mathrm{Gal}_k \rightarrow 1$$ is exact as abstract groups and the map $π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X)$ is a topological embedding. On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_14015 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Fundamental Exact Sequence for the Pro-Étale Fundamental Group Lara, Marcin Algebraic Geometry Number Theory 14F35, 14F20, 14D10, 20E06 The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual étale fundamental group $π_1^{\mathrm{et}}$ defined in SGA1 and the more general group defined in SGA3. It controls local systems in the pro-étale topology and leads to an interesting class of "geometric covers" of schemes, generalizing finite étale covers. We prove the homotopy exact sequence over a field for the pro-étale fundamental group of a geometrically connected scheme $X$ of finite type over a field $k$, i.e. that the sequence $$1 \rightarrow π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X) \rightarrow \mathrm{Gal}_k \rightarrow 1$$ is exact as abstract groups and the map $π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X)$ is a topological embedding. On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group. |
| title | Fundamental Exact Sequence for the Pro-Étale Fundamental Group |
| topic | Algebraic Geometry Number Theory 14F35, 14F20, 14D10, 20E06 |
| url | https://arxiv.org/abs/1910.14015 |