Descente par Frobenius Explicite pour les †-Modules

In his theory of † -modules, P. Berthelot has proved a general theorem of Frobenius descent. On the other hand, Christol and others have also proved various statements about weak Frobenius structures on an annulus of the rigid projective line. It is the aim of this paper to give some explicit and global formulas for Frobenius descent for † -modules. It requires us to build a new kind of differential operator which represents Dworkʹs ψ operator. The construction uses mainly Taylor series properties. It is possible to derive from this some new proofs of Christolʹs theorems; it is also useful for computations in characteristicplike those arising in the construction of Cartierʹs isomorphism or Cartierʹs operator.