On the Swan conductor in positive characteristic

Let X be a proper smooth surface over an algebraically closed field of positive characteristic and U be a complement of a simple normal crossing divisor. For a smooth l-adic sheaf Ƒ on U, Deligne proved a formula calculating the Euler characteristic of Ƒ by local invariants. Kato gave another formula for the Euler characteristic in case where Ƒ is of rank 1, using class field theory. In this paper, we show that local terms of these formulas coincide, admitting certain conjectures for vanishing cycles.