On the refined ramification filtrations in the equal characteristic case

Abstract

Let $k$ be a complete discrete valuation field of equal characteristic $p>0$. Using the tools of $p$-adic differential modules, we define refined Artin and Swan conductors for a representation of the absolute Galois group $G_k$ with finite local monodromy; this leads to a description of the subquotients of the ramification filtration on $G_k$. We prove that our definition of the refined Swan conductors coincides with that given by Saito, which uses étale cohomology. We also study its relation with the toroidal variation of Swan conductors.