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

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.