Let $\mathfrak{o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and $\mathfrak{X}_{0}$ a smooth formal $\mathfrak{o}$-scheme. Let $\mathfrak{X}\rightarrow \mathfrak{X}_{0}$ be an admissible blow-up. In the first part, we introduce sheaves of differential operators $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ on $\mathfrak{X}$, for every sufficiently large positive integer $k$, generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme $\mathfrak{X}_{0}$. The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf $\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves $\mathscr{D}_{\mathfrak{X},\infty }$, over all admissible blow-ups $\mathfrak{X}$, is a sheaf $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ on the Zariski–Riemann space of $\mathfrak{X}_{0}$, which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$, $\mathscr{D}_{\mathfrak{X},\infty }$, and $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$.

中文翻译：

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形式方案上微分算子的算术结构

让$\mathfrak{o}$是一个完全离散的混合特征估值环$(0,p)$和$\mathfrak{X}_{0}$流畅的正式$\mathfrak{o}$-方案。让$\mathfrak{X}\rightarrow \mathfrak{X}_{0}$是一个可接受的爆炸。在第一部分，我们介绍了微分算子的滑轮$\mathscr{D}_{\mathfrak{X},k}^{\dagger }$在$\mathfrak{X}$, 对于每个足够大的正整数$k$, 在平滑形式方案上推广 Berthelot 算术微分算子$\mathfrak{X}_{0}$. 这些滑轮的连贯性和其他几个基本特性得到了证明。第二部分，我们研究了射影极限层。$\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$并介绍其可接纳模的阿贝尔范畴。滑轮的感应极限$\mathscr{D}_{\mathfrak{X},\infty }$, 在所有允许的爆炸中$\mathfrak{X}$, 是一捆$\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$在 Zariski-Riemann 空间$\mathfrak{X}_{0}$，这产生了一个可接纳模块的阿贝尔范畴。定理 A 和 B 的类似物被证明在这些设置中的每一个中都成立，也就是说，对于$\mathscr{D}_{\mathfrak{X},k}^{\dagger }$,$\mathscr{D}_{\mathfrak{X},\infty }$， 和$\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$.