On the $v$-Picard group of Stein spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916370383372288 |
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| author | Ertl, Veronika Gilles, Sally Nizioł, Wiesława |
| author_facet | Ertl, Veronika Gilles, Sally Nizioł, Wiesława |
| contents | We study the image of the Hodge-Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $Z_p$-module that maps, via the Bloch-Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit discs of dimension strictly greater than $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14649 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $v$-Picard group of Stein spaces Ertl, Veronika Gilles, Sally Nizioł, Wiesława Algebraic Geometry Number Theory We study the image of the Hodge-Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $Z_p$-module that maps, via the Bloch-Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit discs of dimension strictly greater than $1$. |
| title | On the $v$-Picard group of Stein spaces |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2312.14649 |