Line bundles on rigid spaces in the v-topology

• Published in: Forum of mathematics. Sigma Vol. 10
• Main Author: Heuer, Ben
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.01.2022
• Subjects: Correspondence; Diamonds; Logarithms; Number Theory; Topology; 14F20; 14C22; 14G45; 11G99
• ISSN: 2050-5094; 2050-5094
• DOI: 10.1017/fms.2022.72

For a smooth rigid space X over a perfectoid field extension K of $\mathbb {Q}_p$ , we investigate how the v-Picard group of the associated diamond $X^{\diamondsuit }$ differs from the analytic Picard group of X. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence $$\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*}$$ We deduce some analyticity criteria which have applications to p-adic modular forms. For algebraically closed K, we show that the sequence is also right-exact if X is proper or one-dimensional. In contrast, we show that, for the affine space $\mathbb {A}^n$ , the image of the Hodge–Tate logarithm consists precisely of the closed differentials. It follows that, up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we use this to construct the p-adic Simpson correspondence of rank one.