Inductive system coherence for logarithmic arithmetic $\mathcal{D}$-modules, stability for cohomology operations

Abstract

Let $\mathcal{V}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $\mathcal{P}$ be a smooth, quasi-compact, separated formal scheme over $\mathcal{V}$, $\mathcal{Z}$ be a strict normal crossing divisor of $\mathcal{P}$ and ${\mathcal{P}}^{♯}:=(\mathcal{P},\mathcal{Z})$ the induced smooth formal log-scheme over $\mathcal{V}$.

In Berthelot's theory of arithmetic $\mathcal{D}$-modules, we work with the inductive system of sheaves of rings ${\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(\bullet )}:=({\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(m)}{)}_{m\in \mathbb{N}}$, where ${\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(m)}$ is the $p$-adic completion of the ring of differential operators of level $m$ over ${\mathcal{P}}^{♯}$. Moreover, he introduced the sheaf ${\mathcal{D}}_{{\mathcal{P}}^{♯},\mathbb{Q}}^{†}:={\underrightarrow{\lim }}_m{\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{(m)}{\otimes }_{\mathbb{Z}}\mathbb{Q}$ of differential operators over ${\mathcal{P}}^{♯}$ of finite level.

In this paper, we define the notion of (over)coherence for complexes of ${\hat{\mathcal{D}}}_{{\mathcal{P}}^{♯}}^{\bullet }$-modules. In this inductive system context, we prove some classical properties including that of Berthelot–Kashiwara's theorem. Moreover, when $\mathcal{Z}$ is empty, we check this notion is compatible to that already know of (over)coherence for complexes of ${\mathcal{D}}_{\mathcal{P},\mathbb{Q}}^{†}$-modules.