Non-commutative intersection theory and unipotent Deligne-Milnor formula

We apply methods of derived and non-commutative algebraic geometry to understand intersection theoretic phenomena on arithmetic schemes. Specifically, we categorify Bloch's intersection number (in the formulation provided by Kato--Saito). Combining this with To\"en--Vezzosi's non-commutative Chern character, we obtain a generalization of Bloch conductor conjecture in several new cases, including the unipotent Deligne--Milnor formula.