Reconstruction theorems for coadmissible D-cap-modules

TL;DR

This paper establishes a Riemann-Hilbert correspondence for coadmissible D-cap-modules, showing they can be reconstructed from their solutions, and demonstrates the full faithfulness of certain solution functors in this context.

Contribution

It introduces a reconstruction theorem for coadmissible D-cap-modules and proves the full faithfulness of modified solution and de Rham functors for these modules.

Findings

Reconstruction of coadmissible D-cap-modules from solutions.

Full faithfulness of modified solution and de Rham functors.

Explicit computation of Galois cohomology of a decompletion of B_{dR}^{+}.

Abstract

We prove a Riemann-Hilbert correspondence for Ardakov-Wadsley's coadmissible D-cap-modules and, more generally, for Bode's $\mathcal{C}$-complexes. More precisely, we show that any given $\mathcal{C}$-complex can be reconstructed out of its solutions. As a corollary, we find that slight modifications of the solution and de Rham functors introduced by the author are fully faithful on $\mathcal{C}$-complexes and, in particular, on coadmissible D-cap-modules. One of the many steps of our proof is the explicit computation of the continuous Galois cohomology of a certain decompletion of $B_{dR}^{+}$ which we call the positive overconvergent de Rham period ring.