On the $v$-Picard group of Stein spaces

Abstract

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 $1$.

arXiv commentFinal version; to appear in IMRN

References in corpus (10)

• Relative p-adic Hodge theory, II: Imperfect period rings
• Line bundles on perfectoid covers: case of good reduction
• Line bundles on rigid spaces in the $v$-topology
• Diamantine Picard functors of rigid spaces
• The Primitive Comparison Theorem in characteristic $p$
• Cohomologie analytique des arrangements d'hyperplans
• A $p$-adic Simpson correspondence for smooth proper rigid varieties
• On the cohomology of p-adic analytic spaces, II: The $C_{\mathrm{st}}$-conjecture
• Rational $p$-adic Hodge theory for rigid-analytic varieties
• Homotopy Classification of Line Bundles Over Rigid Analytic Varieties