Categorification of the localized intersection product and Bloch conductor formula
Dario Beraldo, Massimo Pippi
TL;DR
The paper develops a categorified framework for the localized intersection product on arithmetic schemes and uses it to prove Bloch's conductor formula and its generalized version. Central to the construction is the convolution monoidal dg-category $\mathsf{B}$ on the derived groupoid $G=s\times_S s$, and its Drinfeld cocenter $\operatorname{HH}(\mathsf{B}/A)$, together with an $\,\ell$-adic realization that decategorifies to numerical invariants. A key innovation is a filtration of the cocenter via a cyclic-bar construction and a system of coherent higher homotopies that yield a dg-functor $\oint$ collapsing the categorified data to the classical Bloch numbers; this provides a principled reason for why Bloch-type formulas hold in mixed characteristic. The work also clarifies the pure-characteristic simplifications, where the convolution structure becomes symmetric and a retraction of the cocenter exists, linking noncommutative Chern characters to classical invariants. Overall, the paper supplies a robust categorical mechanism to translate between noncommutative/homotopical data and arithmetic intersection theory, enabling the deduction of gBCC from its categorical counterpart and offering new tools for understanding localized intersection products in arithmetic geometry.
Abstract
We categorify the localized intersection product on arithmetic schemes defined by Kato--Saito in \cite{katosaito04}. As an application, we prove a generalization of Bloch conductor conjecture.