Relative tensor triangular Chow groups for coherent algebras

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.Comment: 39 pages; all comments welcom

A guide to tensor-triangular classification

“…Moving away from this preoccupation, some first 'geometric' results were established in [Bal07], like the decomposition of an object associated to a decomposition of its support, and applications to filtrations of K by (co)dimension of support. These ideas naturally led to tensor-triangular Chow groups in [Bal13] and further improvements by Klein [Kle16] and Belmans-Klein [BK17], using the already mentioned relative tt-geometry of [Ste13]. 8.6.…”