Line bundles on rigid spaces in the<i>v</i>-topology

Abstract For a smooth rigid space X over perfectoid field extension K of $\mathbb {Q}_p$ , we investigate how the v -Picard group associated diamond $X^{\diamondsuit }$ differs from analytic Picard . To this end, construct 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. algebraically closed show that is also right-exact if proper or one-dimensional. In contrast, that, for affine {A}^n$ image Hodge–Tate logarithm consists precisely differentials. It follows up splitting, -line bundles may be interpreted as Higgs bundles. use Simpson correspondence rank one.