Relative tensor triangular Chow groups, singular varieties and localization

TL;DR

This paper generalizes tensor triangular Chow groups to broader categories, enabling the recovery of Chow groups for singular varieties and establishing localization sequences akin to classical algebraic geometry results.

Contribution

It extends Balmer's tensor triangular Chow groups to categories with only an action by a tensor triangulated category, not necessarily monoidal, allowing applications to singular varieties.

Findings

Recovered Chow groups of singular algebraic varieties from homotopy categories.

Constructed localization sequences for open subsets in the Balmer spectrum.

Extended the framework to categories with only an action, not a monoidal structure.

Abstract

We extend the scope of Balmer's tensor triangular Chow groups to compactly generated triangulated categories $\mathcal{K}$ that only admit an action by a compactly-rigidly generated tensor triangulated category $\mathcal{T}$ as opposed to having a compatible monoidal structure themselves. The additional flexibility allows us to recover the Chow groups of a possibly singular algebraic variety $X$ from the homotopy category of quasi-coherent injective sheaves on $X$. We are also able to construct localization sequences associated to restricting to an open subset of $\mathrm{Spc}(\mathcal{T}^c)$, the Balmer spectrum of the subcategory of compact objects $\mathcal{T}^c \subset \mathcal{T}$. This should be viewed in analogy to the exact sequences for the cycle and Chow groups of an algebraic variety associated to the restriction to an open subset.