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Differential coherence of overconvergent function algebras. (Cohérence différentielle des algèbres de fonctions surconvergentes.) (French. Abridged English version) Zbl 0871.14014
Let \(({\mathcal V}, {\mathfrak m})\) be a complete discrete valuation ring of unequal characteristics \((0,p)\) and \({\mathcal X}\) a smooth formal \({\mathcal V}\)-scheme. Let \(X\) be the reduction of \({\mathcal X} \bmod {\mathfrak m}\). The author proves that, for any divisor \(Z\subset X\), the sheaf \({\mathcal O}_{{\mathcal X}, \mathbb{Q}} (^† Z)\) of functions with overconvergent singularities along \(Z\) is coherent as a \({\mathcal D}_{{\mathcal X}, \mathbb{Q}}\)-module. The proof depends essentially on the weak resolution theorem of A. J. de Jong [“Smoothness, semi-stability and alterations” (to appear)]. This result, together with the coherence theorem for proper direct images previously stated by the author, reduces the general case to the normal crossing case. When \(Z\) is a normal crossing divisor, the result follows by the same computation as in the complex analytic context.

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14H20 Singularities of curves, local rings
14G20 Local ground fields in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)