Logarithmic geometry beyond fs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917843463831552 |
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| author | Achinger, Piotr Hübner, Katharina Lara, Marcin Stix, Jakob |
| author_facet | Achinger, Piotr Hübner, Katharina Lara, Marcin Stix, Jakob |
| contents | We develop the foundations of logarithmic structures beyond the standard finiteness conditions. The motivation is the study of semistable models over general valuation rings. The key new notion is that of a morphism of finite presentation up to saturation (sfp), which is one that is qcqs and which is locally isomorphic to the saturated base change of a finitely presented morphism between fs log schemes. As in the case of schemes, sfp maps can (locally on the base) be approximated by maps between fs log schemes of finite type over $\mathbb{Z}$. Based on sfp maps, we define smooth, étale, and Kummer étale maps. Importantly, the maps of schemes underlying such maps are no longer of finite type in general, though surprisingly they are if the base is the spectrum of a valuation ring with algebraically closed field of fractions. These foundations allow us to extend beyond the fs case the theory of the Kummer étale site and of the Kummer étale fundamental group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Logarithmic geometry beyond fs Achinger, Piotr Hübner, Katharina Lara, Marcin Stix, Jakob Algebraic Geometry 14A21 (Primary), 14F35 (Secondary) We develop the foundations of logarithmic structures beyond the standard finiteness conditions. The motivation is the study of semistable models over general valuation rings. The key new notion is that of a morphism of finite presentation up to saturation (sfp), which is one that is qcqs and which is locally isomorphic to the saturated base change of a finitely presented morphism between fs log schemes. As in the case of schemes, sfp maps can (locally on the base) be approximated by maps between fs log schemes of finite type over $\mathbb{Z}$. Based on sfp maps, we define smooth, étale, and Kummer étale maps. Importantly, the maps of schemes underlying such maps are no longer of finite type in general, though surprisingly they are if the base is the spectrum of a valuation ring with algebraically closed field of fractions. These foundations allow us to extend beyond the fs case the theory of the Kummer étale site and of the Kummer étale fundamental group. |
| title | Logarithmic geometry beyond fs |
| topic | Algebraic Geometry 14A21 (Primary), 14F35 (Secondary) |
| url | https://arxiv.org/abs/2411.13662 |