Arithmetic structures for differential operators on formal schemes

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 ${\mathcal{D}}_{\mathfrak{X},k}^{†}$ on $\mathfrak{X}$, for every integer $k\ge k_{\mathfrak{X}}$, where $k_{\mathfrak{X}}$ depends on the blow-up morphism $\mathfrak{X}\to {\mathfrak{X}}_0$. 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 ${\mathscr D}{{\mathfrak X},\infty} = \varprojlim_k {\mathscr D}^\dagger{{\mathfrak X},k}$andso-calledcoadmissiblemodulesfor${\mathscr D}{{\mathfrak X},\infty}$\mathrm{.}Theinductivelimitofthesheaves${\mathscr D}{{\mathfrak X},\infty}$,overalladmissibleblow-ups${\mathfrak X}$of${\mathfrak X}0$, gives rise to a sheaf ${\mathcal{D}}_{⟨{\mathfrak{X}}_0⟩}$ 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${\mathscr D}{\langle {\mathfrak X}_0\rangle}$.