  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.
