On the \(v\)-Picard group of Stein spaces

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 differe...

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Bibliographic Details
Published in:arXiv.org 2024-08
Main Authors: Ertl, Veronika, Gilles, Sally, Nizioł, Wiesława
Format: Article
Language:English
Subjects:
Homology
Quotients
Online Access:Get full text
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Summary: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 \(1\).
ISSN:2331-8422