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

TL;DR

The paper investigates higher analytic vectors in the Lubin–Tate–twisted Fontaine period ring setting, proving that the first derived analytic vectors of the $H_K$-stable subring $\mathbf{B}_e$ are nonzero and computing their analytic cohomology. It also describes the cokernel of a Bloch–Kato–type exponential map restricted to analytic vectors for $\mathbb{Q}_p(n)$ via derived analytic vectors, and links pro-analytic to derived analytic vectors within condensed mathematics for regular LF-spaces. The approach blends $p$-adic Hodge theory, condensed mathematics, and derived functor technology to compare analytic and Galois cohomology through spectral sequences and exact sequences. The results illuminate when analytic vectors suffice to recover Galois cohomology and provide structural descriptions of analytic invariants for period rings, with implications for understanding $H_K$-actions and Lubin–Tate extensions in a condensed setting.

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.

Theorems & Definitions (51)

• Theorem 1
• Theorem 2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 3.1
• proof
• Definition 3.2
• Remark 3.3