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}^\sharp:=({\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 \widehat{{\mathcal D}}^{(\bullet)}_{{\mathcal P}^\sharp}:=(\widehat{{\mathcal D}}^{(m)}_{{\mathcal P}^\sharp})_{m\in\mathbb{N}} , where \widehat{{\mathcal D}}^{(m)}_{{\mathcal P}^\sharp} is the p -adic completion of the ring of differential operators of level m over {\mathcal P}^\sharp . Moreover, he introduced the sheaf {\mathcal D}^\dag_{{\mathcal P}^\sharp,\mathbb{Q}}:= \varinjlim_m\,\widehat{{\mathcal D}}^{(m)}_{{\mathcal P}^\sharp}\otimes_{\mathbb{Z}}\mathbb{Q} of differential operators over {\mathcal P}^\sharp of finite level. In this paper, we define the notion of (over)coherence for complexes of \widehat{{\mathcal D}}^{\bullet}_{{\mathcal P}^\sharp} -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}^\dag_{{\mathcal P},\mathbb{Q}} -modules.