Abstract
Let
{\mathcal X}^\dagger
be a smooth
\dagger
-scheme (in the sense of Meredith) over a complete discrete valuation ring
(V, {\mathfrak m})
of unequal characteristics
(0,p)
and let
{\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}
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 we prove that locally there is an
{\mathcal O}_{{\mathcal X}^\dagger}
-isomorphism between the sections of
{\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}
and the overconvergent total symbols, and we deduce a cohomological triviality property.