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Geometric interpretation on the projective space of coherent $$A_ N(K)^†$$-modules. (Interprétation géométrique sur l’espace projectif des $$A_ N(K)^†$$-modules cohérents.) (French) Zbl 0872.14010
Let $$\mathcal V$$ be a complete discrete valuation ring of unequal characteristics $$(0,p)$$, $$K$$ its fraction field and $$X$$ (resp. $$\mathcal X$$) the projective (resp. the formal projective) space over $$\mathcal V$$. In this note, the author proves that the global sections of the sheaf $${\mathcal D}^\dagger_{{\mathcal X},\mathbf Q}(\infty)$$ of differential operators with overconvergent singularities at infinity, defined by P. Berthelot in Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 2, 185-272 (1996), coincide with the weak completion $$A_N(K)^\dagger$$ of the Weyl algebra, i.e. the sections over the affine space of the sheaf $${\mathcal D}^\dagger_{X^\dagger}\otimes \mathbb{Q}$$ introduced by Z. Mebkhout and L. Narváez-Macarro Notes Math. 1454, 267-308 (1990; Zbl 0727.14011), where $$X^\dagger$$ is the weak completion of $$X$$. As a consequence, she obtains an equivalence between the category of coherent $$A_N(K)^\dagger$$-modules and the category of coherent $${\mathcal D}^\dagger_{{\mathcal X},\mathbf Q}(\infty)$$-modules.
[Reviewer’s remark: the author calls the ring $$R:=\bigl\{\displaystyle \sum_{l,k} a_{l,k} x^l \partial_x^k/k!\mid \exists C,\eta< 1$$, $$a_{l,k} < C \eta^{l+k}\bigr\}$$ “weak completion of the Weyl algebra”. In the reviewer’s opinion, it would be more appropriate to reserve this name for the ring $$\bigl\{\displaystyle \sum_{l,k} a_{l,k} x^l \partial_x^k\quad \exists C,\eta< 1$$, $$a_{l,k} < C \eta^{l+k}\bigr\}$$ and the ring $$R$$ would be called “weak completion of the ring of differential operators”].

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials