On smooth adic spaces over \(\mathbb{B}_{\mathrm{dR}}^+\) and sheafified \(p\)-adic Riemann--Hilbert correspondence

Abstract

Inspired by Heuer's paper, we consider adic spaces over a de Rham period ring of a perfectoid Tate--Huber ring and their sheafified Riemann--Hilbert correspondence. We will prove that for any smooth adic space \(X\) over \(\mathbb{B}_{\mathrm{dR},\alpha}^+(K,K^+)\), there is a canonical sheaf isomorphism \[R^1\nu_*\left(\mathrm{GL}_r({\mathbb{B}_{\mathrm{dR}}^+}_{,\bar{X}}/t^\alpha)\right)\cong t-\mathrm{MIC}_r(X).\] Moreover, we will define \(v\)-prestacks of \(\mathbb{B}_{\mathrm{dR}}^+\)-local systems and \(t\)-connections, and prove that they are small \(v\)-stacks.