The total symbol theorem of a \(p\)-adic differential operator. (Le théorème du symbole total d’un opérateur différentiel \(p\)-adique.) (French) Zbl 1213.14042

Mebkhout, Zoghman; Narváez Macarro, Luis

Rev. Mat. Iberoam. 26, No. 3, 825-859 (2010).

This article continues earlier work [in: \(p\)-adic analysis, Proc. Int. Conf., Trento 1989, Lect. Notes Math. 1454, 267–308 (1990; Zbl 0727.14011)] of the authors on finiteness properties of rings of \(p\)-adic differential operators on weak formal schemes (the authors write: \({\dagger}\)-adic schemes). As the authors write, most of the results have been obtained in the late 80’s without having been published, but due to their central role their publication now becomes necessary.
Let \({\mathcal X}^{\dagger}\) denote a smooth weak formal scheme over a complete discrete valuation ring \((V,{\mathfrak m})\) of unequal characteristic \((0,p)\) and let \({\mathcal D}_{{\mathcal X}^{\dagger}/V}^{\dagger}\) be the sheaf of \(V\)-linear endomorphisms of \({\mathcal O}_{{\mathcal X}^{\dagger}}\) whose reduction modulo \({\mathfrak m}^s\) is a linear differential operator of order bounded by an affine function in \(s\). In this paper it is proved that locally there is an \({\mathcal O}_{{\mathcal X}^{\dagger}}\)-isomorphism between the sections of \({\mathcal D}_{{\mathcal X}^{\dagger}/V}^{\dagger}\) and the overconvergent symbols.
More precisely, let \(A^{\dagger}\) be a weakly complete \(V\)-algebra with coordinate functions \(x_1,\ldots,x_n\) such that their differentials form a basis of the \(A^{\dagger}\)-module of separated differetial forms. Let \(P\) be a differential operator of the ring \(D^{\dagger}_{A^{\dagger}/V}\). For \(\alpha\in {\mathbb N}^{n}\) define \(a_{\alpha}\in A^{\dagger}\) through \[ a_{\alpha}=\sum_{0\leq\beta\leq\alpha}{\alpha\choose\beta}(-x)^{\beta}P(x^{\alpha-\beta}). \] Then it is shown that \[ P=\sum_{\alpha}a_{\alpha}\Delta_x^{\alpha}. \] Moreover, the assignment \[ P\mapsto \sigma_P(x,\xi):=\sum_{\alpha}a_{\alpha}\xi^{\alpha} \] is an isomorphism of \(A^{\dagger}\)-modules \[ D^{\dagger}_{A^{\dagger}/V}\cong A^{\dagger}[\xi_1,\ldots,\xi_n]^{\dagger}. \]

Reviewer: Elmar Große-Klönne (Berlin)

MSC:

14F30	\(p\)-adic cohomology, crystalline cohomology
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Keywords:

weak formal scheme; dagger adic scheme; weakly complete algebra; overconvergent symbol; \(p\)-adic differential operator

Citations:

Zbl 0727.14011

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