Non-commutative intersection theory and unipotent Deligne-Milnor formula

TL;DR

This paper employs derived and non-commutative algebraic geometry techniques to generalize intersection theory and Bloch's conductor conjecture, leading to new insights such as the unipotent Deligne--Milnor formula.

Contribution

It introduces a categorification of Bloch's intersection number and extends the Bloch conductor conjecture using non-commutative methods, including the non-commutative Chern character.

Findings

Generalization of Bloch conductor conjecture in new cases

Categorification of intersection numbers on arithmetic schemes

Derivation of the unipotent Deligne--Milnor formula

Abstract

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.