Mathematical Research Letters
Volume 12 (2005)
Frobenius modules and de Jong’s theorem
Pages: 303 – 320
Let $k$ be an algebraically closed field of characteristic $p >0$. A theorem of de~Jong shows that morphisms of modules over $W(k) \llbracket t \rrbracket$ with Frobenius and connection structure descend from the completion of $W(k)((t))$. A careful reading of de~Jong’s proof suggests the possibility that an analogous theorem holds for modules with only a Frobenius structure. We show that this analogue holds in one natural formulation, but fails in a stronger formulation in which $W(k) \llbracket t \rrbracket$ is replaced by $W(k \llbracket t \rrbracket)$.