Relative tensor triangular Chow groups for coherent algebras

TL;DR

This paper extends tensor triangular Chow groups to noncommutative algebras over schemes, providing new descriptions and concrete examples, and linking these invariants to classical algebraic structures.

Contribution

It introduces a framework for relative tensor triangular Chow groups for noncommutative algebras and offers explicit computations and relations to classical invariants.

Findings

Recovered tensor triangular Chow groups for commutative cases

Provided concrete descriptions for group algebras and hereditary orders

Linked Chow groups to classical ideal class groups

Abstract

We apply the machinery of relative tensor triangular Chow groups to the action of the derived category of quasi-coherent sheaves on a noetherian scheme $X$ on the derived category of quasi-coherent $\mathcal{A}$-modules, where $\mathcal{A}$ is a (not necessarily commutative) quasi-coherent $\mathcal{O}_X$-algebra. When $\mathcal{A}$ is commutative and coherent, we recover the tensor triangular Chow groups of the relative Spec of $\mathcal{A}$. We also obtain concrete descriptions for integral group algebras and hereditary orders over curves, and we investigate the relation of these invariants to the classical ideal class group of an order. An important tool for these computations is a new description of relative tensor triangular Chow groups as the image of a map in the K-theoretic localization sequence associated to a certain Verdier localization.