Logarithmic decomposition of connections on a relatively punctured disk

Abstract

Let $R=C[[t]]$ be the ring of power series over an algebraically closed field $C$ of characteristic zero. We show that each connection on a finite flat $R((x))$-module is the sum of a regular singular connection and a diagonalizable $R((x))$-linear endomorphism when it admits a Turrittin-Levelt-Jordan form over $R((x))$. This decomposition is compatible with the limit of the logarithmic decompositions of the connections obtained by the reduction modulo $t^{k}$ of a given connection.