Logarithmic decomposition of connections on a relatively punctured disk

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