## 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.

### MSC:

 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)