Systèmes inductifs cohérents de D-modules arithmétiques logarithmiques, stabilité par opérations cohomologiques

Let $V$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $P$ be a smooth, quasi-compact, separated formal scheme over $V$, $Z$ be a strict normal crossing divisor of $P$ and $\PP \sharp:= (\PP,\ZZ)$ the induced smooth formal log-scheme over $\V$. In Berthelot's theory of arithmetic $D$-modules, we work with the inductive system of sheaves of rings $\smash{\widehat{D}} P \sharp (\bullet) : = (\smash{\widehat{D}} P\sharp (m))m\in \N$, where $\smash{\widehat{D}} P\sharp (m)$ is the $p$-adic completion of the ring of differential operators of level $m$ over $P\sharp$. Moreover, he introduced the sheaf $D \dag P \sharp,\{Q}:=\underset{\underset{m}{\longrightarrow}}{\lim} \smash{\widehat{D}} P \sharp (m) \otimes \{Z }\{Q}$ of differential operators over $P \sharp$ of finite level. In this paper, we define the notion of (over)coherence for complexes of $\smash{\widehat{D}} P \sharp (\bullet) $-modules. In this inductive system context, we prove some classical properties including that of Berthelot-Kashiwara's theorem. Moreover, when $\ZZ$ is empty, we check this notion is compatible to that already known of (over)coherence for complexes of $D \dag P,\{Q}$-modules.

2010 Mathematics Subject Classification: 14F30

Keywords and Phrases: Arithmetic D-modules, p-adic cohomology, de Rham cohomology

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