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We prove the functoriality for proper push-forward of the characteristiccycles of constructible complexes by morphisms of smooth projective schemesover a perfect field, under the assumption that the direct image of thesingular support has the dimension at most that of the target of the morphism.The functoriality is deduced from a conductor formula which is a special casefor morphisms to curves. The conductor formula in the constant coefficient casegives the geometric case of a formula conjectured by Bloch.

Characteristic cycles and the conductor of direct image