On the higher analytic vectors of $\mathbf{B}_e$

TL;DR

This paper investigates the structure of higher analytic vectors in Fontaine's period ring _e, establishing their non-triviality, computing their cohomology, and relating them to exponential maps in p-adic Hodge theory.

Contribution

It proves the non-vanishing of first derived analytic vectors in _e and describes their cohomology, connecting pro-analytic and derived analytic vectors in condensed mathematics.

Findings

First derived analytic vectors of _e are non-zero.

Computed the analytic cohomology of these vectors.

Described the cokernel of a variant of the Bloch-Kato exponential map.

Abstract

We prove that the first derived analytic vectors of the subring of Fontaine's period ring $\mathbf{B}_e$ stable under the kernel of the cyclotomic character are non-zero. Subsequently we compute their analytic cohomology. We also give a description of the cokernel of the restriction of a variant of the Bloch-Kato exponential map for $\mathbb{Q}_p(n)$ to analytic vectors in terms of derived analytic vectors. In order to achieve the above, we relate pro-analytic vectors with derived analytic vectors in condensed mathematics for regular LF-spaces.