Systèmes inductifs cohérents de ${\mathcal D}$-modules arithmétiques logarithmiques, stabilité par opérations cohomologiques

Caro, Daniel

Documenta mathematica, Tome 21 (2016), pp. 1515-1606.

Résumé

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 ${\mathcal V}$. \par 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. \par 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.

Détail

Classification : 14F30, 14F10
Keywords: arithmetic \({\mathcal D}\)-modules, \(p\)-adic cohomology, de Rham cohomology