AI 中文总结
本文针对p进域的p进李扩张，研究de Rham周期环的前解析向量子环，建立其与特定形式幂级数环的伽罗瓦等变同构条件，并将所得Sen算子提升结果应用于正则联络构造与伽罗瓦上同调计算。
AI 中文摘要
设$K_\infty/K$为p进域$K$的p进李扩张，本文研究de Rham周期环$\mathbf{B}_{\mathrm{dR}}^+(K_\infty)$中的前解析向量子环，证明该前解析子环与形式幂级数环$\widehat{K}_{\infty}^{\mathrm{la}} [[t_{K_\infty}]]$存在伽罗瓦等变同构当且仅当$K_\infty$满足特定可定向性条件，即$\widehat{K}_\infty$级Sen算子存在伽罗瓦等变$\mathbf{B}_{\mathrm{dR}}^+$-提升；关键输入是$\widehat{K}_\infty$表示的高阶局部解析向量消失，作为应用，本文证明提升后的Sen算子在$\mathbf{B}_{\mathrm{dR}}^+$表示的前解析向量上诱导正则联络，可用于计算伽罗瓦上同调。
英文摘要
Let $K_\infty/K$ be a $p$-adic Lie extension of a $p$-adic field $K$. We study the subring of pro-analytic vectors in the de Rham period ring $\mathbf{B}_{\mathrm{dR}}^+(K_\infty)$. We show that the pro-analytic subring admits a Galois-equivariant isomorphism with a formal power series ring $\widehat{K}_{\infty}^{\mathrm{la}} [[t_{K_\infty}]]$ if and only if $K_\infty$ satisfies a certain orientability condition, which says that the $\widehat{K}_\infty$-level Sen operator admits a Galois-equivariant $\mathbf{B}_{\mathrm{dR}}^+$-lift. A key input is the vanishing of higher locally analytic vectors of $\widehat{K}_\infty$-representations. As an application, we show that the lifted Sen operator induces regular connections on pro-analytic vectors of $\mathbf{B}_{\mathrm{dR}}^+$-representations, and can be used to compute Galois cohomology.
