Inductive system coherence for logarithmic arithmetic \({\mathcal D}\)-modules, stability for cohomology operations
Summary: 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.
scientific article; zbMATH DE number 6674985