Cohérence différentielle des F-isocristaux unités

Let $\text{V}$ be a mixed characteristic complete discrete valuation ring, $\text{P}$ a smooth formal scheme over $\text{V}$, P its special fiber, X a smooth subscheme of P, T a divisor in P such that T_X=T∩X is a divisor in X and $\text{D}{}^2{}_{\mathrm{<mtext>P</mtext>}}$ the weak completion of the sheaf of differential operators on $\text{P}$. We prove that the unit-root F-isocrystals on X?T_X overconvergent along T_X are coherent over $\text{D}{}^2{}_{\mathrm{<mtext>P</mtext><mtext>,</mtext><mtext>Q</mtext>}}$. To cite this article: D. Caro, C. R. Acad. Sci. Paris, Ser. I 338 (2004).