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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”].

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