Relative tensor triangular Chow groups for coherent algebras

Abstract

We apply the machinery of relative tensor triangular Chow groups to the action of D(Qcoh(X)), the derived category of quasi-coherent sheaves on a noetherian scheme X, on the derived category of quasi-coherent A-modules D(Qcoh(A)), where A is a (not necessarily commutative) coherent OX-algebra. When A is commutative, we recover the tensor triangular Chow groups of Spec(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.