论文标题
在非架构的几何形状中散布霍奇过滤
Spreading out the Hodge filtration in non-archimedean geometry
论文作者
论文摘要
当前文本的目的是通过正式完成研究非架构分析衍生的DE RHAM的共同体。我们的方法的灵感来自于\ cite {gaitsgory_study_ii}中提供的正常锥体的变形。更具体地说，给定（派生的）$ k $ - 分析空间的形态$ f \ colon x \，我们构建了与$ f $相关的正常锥体的\ emph {non-archimedean变形。后者可以被认为是$ \ mathbf a^1_k $ -parametrized变形的变形，其光纤为$ 1 \ in \ Mathbf a^1_k $与$ f $的正式完成和$ 0 \ in \ mathbf a^1_k $与与$ f $相关的（派生的）正常锥体的正式完成和纤维相吻合。我们进一步表明，这种变形可以赋予自然过滤，从而在（已移动的）分析切线束上散布通常的Hodge过滤到正式的完成。在$ f $是（派生的）$ k $ affinoid空间之间的本地完整的交点形态的情况下，这种过滤与$ i $ aidic的过滤一致。在此过程中，我们发展了（Ind-Inf） - $ K $分析空间或换句话说，$ k $ - 分析正式模量问题。
The goal of the current text is to study non-archimedean analytic derived de Rham cohomology by means of formal completions. Our approach is inspired by the deformation to the normal cone provided in \cite{Gaitsgory_Study_II}. More specifically, given a morphism $f \colon X \to Y$ of (derived) $k$-analytic spaces we construct the \emph{non-archimedean deformation to the normal cone} associated to $f$. The latter can be thought as an $\mathbf A^1_k$-parametrized deformation whose fiber at $1 \in \mathbf A^1_k$ coincides with the formal completion of $f$ and the fiber at $0 \in \mathbf A^1_k$ with the (derived) normal cone associated to $f$. We further show that such deformation can be endowed with a natural filtration which spreads out the usual Hodge filtration on the (completed shifted) analytic tangent bundle to the formal completion. Such filtration agrees with the $I$-adic filtration in the case where $f$ is a locally complete intersection morphism between (derived) $k$-affinoid spaces. Along the way we develop the theory of (ind-inf)-$k$-analytic spaces or in other words $k$-analytic formal moduli problems.
