论文标题
a $ p $ - 辅助6函数形式主义在刚性分析的几何形状中
A $p$-Adic 6-Functor Formalism in Rigid-Analytic Geometry
论文作者
论文摘要
我们开发了$ p $ torsionétale滑轮的完整的6函数形式主义。更具体地说，我们使用克劳森（Clausen）的最近开发的凝结数学 - 与伪分子的$ x $相关，将$ x $与伪分子化$π$π$π$π$ \ in $ \ infty $ -caltation $ \ category $ \ caltemation $ \ natercal d^a_ quard（ $ x $ x $ $ \ MATHCAL O^+_ X/π$ -MODULES”。然后，我们在此设置中构建了六个函数$ \ otimes $，$ \ usewise {hom} $，$ f^*$，$ f _*$，$ f _*$，$ f _！$和$ f^！$，并表明它们满足所有预期的兼容性，类似于$ \ ell $ - eLL $ $ $ $ $ $ -ADIC CASE。通过介绍$φ$ - 模块结构并证明了$ p $ - torsion riemann-hilbert通讯的版本，我们将$ \ m i \ mathcal o^+_ x/π$ -sheaves与$ \ Mathbb f_p $ -sheaves相关。作为这种形式主义的特殊情况，我们证明了Poincaré二元性，用于$ \ Mathbb f_p $ - 在刚性分析品种上的生物学。在构建$ \ Mathcal d^a_ \ square（\ Mathcal O^+_ X/π）$的过程中，我们还开发了用于凝结环上的凝结模块的一般血统形式主义。
We develop a full 6-functor formalism for $p$-torsion étale sheaves in rigid-analytic geometry. More concretely, we use the recently developed condensed mathematics by Clausen--Scholze to associate to every small v-stack (e.g. rigid-analytic variety) $X$ with pseudouniformizer $π$ an $\infty$-category $\mathcal D^a_\square(\mathcal O^+_X/π)$ of "derived quasicoherent complete topological $\mathcal O^+_X/π$-modules" on $X$. We then construct the six functors $\otimes$, $\underline{Hom}$, $f^*$, $f_*$, $f_!$ and $f^!$ in this setting and show that they satisfy all the expected compatibilities, similar to the $\ell$-adic case. By introducing $φ$-module structures and proving a version of the $p$-torsion Riemann-Hilbert correspondence we relate $\mathcal O^+_X/π$-sheaves to $\mathbb F_p$-sheaves. As a special case of this formalism we prove Poincaré duality for $\mathbb F_p$-cohomology on rigid-analytic varieties. In the process of constructing $\mathcal D^a_\square(\mathcal O^+_X/π)$ we also develop a general descent formalism for condensed modules over condensed rings.
