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 ${\mathcal{D}}_{\mathfrak{X},k}^{†}$ 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 ${\mathcal{D}}_{\mathfrak{X},\infty }={\underleftarrow{\lim }}_k{\mathcal{D}}_{\mathfrak{X},k}^{†}$ and so-called coadmissible modules for ${\mathcal{D}}_{\mathfrak{X},\infty }$. The inductive limit of the sheaves ${\mathcal{D}}_{\mathfrak{X},\infty }$, over all admissible blow-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 ${\mathcal{D}}_{\mathfrak{X},k}^{†}$, ${\mathcal{D}}_{\mathfrak{X},\infty }$, and ${\mathcal{D}}_{⟨{\mathfrak{X}}_0⟩}$.

Some error corrected and some examples added