Cohomological Milnor formula and Saito's conjecture on characteristic classes

We construct a cohomological characteristic class supported on the non-locally acyclic locus of a separated morphism relatively to a constructible sheaf. We confirm the projective case of Saito's conjecture [Invent.Math.207,597-695 (2017)], namely the cohomological characteristic classes can be computed in terms of the characteristic cycles. As applications of the functorial properties of this class, we prove cohomological analogs of the Milnor formula and the conductor formula for constructible sheaves on (not necessarily smooth) varieties.