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 P♯ := (P,Z) 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 D̂ P♯ := (D̂ P♯ )m∈N, where D̂ (m) P♯ is the p-adic completion of the ring of differential operators of level m over P♯. Moreover, he introduced the sheafD P♯,Q := lim −→ m D̂ (m) P♯ ⊗ZQ of differential operators over P♯ of finite level. In this paper, we define the notion of (over)coherence for complexes of D̂ P♯ -modules. In this inductive system context, we prove some classical properties including that of Berthelot-Kashiwara’s theorem. Moreover, when Z is empty, we check this notion is compatible to that already known of (over)coherence for complexes ofD P,Q-modules. 2010 Mathematics Subject Classification: 14F30