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We apply the machinery of relative tensor triangular Chow groups to theaction 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, werecover the tensor triangular Chow groups of the relative Spec of$\mathcal{A}$. We also obtain concrete descriptions for integral group algebrasand hereditary orders over curves, and we investigate the relation of theseinvariants to the classical ideal class group of an order. An important toolfor these computations is a new description of relative tensor triangular Chowgroups as the image of a map in the K-theoretic localization sequenceassociated to a certain Verdier localization.

Relative tensor triangular Chow groups for coherent algebras