  The Grothendieck-Ogg-Shafarevich formula is generalized to any dimensional
scheme by Abbes-Kato-Saito. In this paper, we introduce two methods of
localization of the characteristic classes for sheaves of rank 1 and compare
them. As a corollary of this comparison, we obtain a refinement of a formula
proved by Abbes-Saito without denominator for a smooth sheaf of rank 1 which is
clean with respect to the boundary.
