Arithmetic structures for differential operators on formal schemes

Abstract

Let ${\mathfrak o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and ${\mathfrak X}_0$ a smooth formal scheme over the formal spectrum of ${\mathfrak o}$. Given an admissible formal blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$ we introduce sheaves of differential operators ${\mathscr D}^\dagger_{{\mathfrak X},k}$ on ${\mathfrak X}$, for every integer $k \ge k_{\mathfrak X}$, where $k_{\mathfrak X}$ depends on the blow-up morphism ${\mathfrak X}\rightarrow {\mathfrak X}_0$. This generalizes Berthelot's construction of sheaves of arit hmetic differential operators on ${\mathfrak X}_0$. The coherence of these sheaves and several other basic properties are proven. In the second part we study the projective limit sheaf ${\mathscr D}_{{\mathfrak X},\infty} = \varprojlim_k {\mathscr D}^\dagger_{{\mathfrak X},k}$ and so-called coadmissible modules for ${\mathscr D}_{{\mathfrak X},\infty}$. The inductive limit of the sheaves ${\mathscr D}_{{\mathfrak X},\infty}$, over all admissible blow-ups ${\mathfrak X}$ of ${\mathfrak X}_0$, gives rise to a sheaf ${\mathscr D}_{\langle {\mathfrak X}_0 \rangle}$ on the Zariski-Riemann space of ${\mathfrak X}_0$. Analogues of Theorems A and B are shown to hold in each of these settings, i.e., for ${\mathscr D}^\dagger_{{\mathfrak X},k}$, ${\mathscr D}_{{\mathfrak X},\infty}$, and ${\mathscr D}_{\langle {\mathfrak X}_0\rangle}$.