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Intersection products for tensor triangular Chow groups

We show that under favorable circumstances, one can construct an intersection product on the Chow groups of a tensor triangulated category $\mathcal{T}$ (as defined by Balmer) which generalizes the usual intersection product on a non-singular algebraic variety. Our construction depends on the choice of an algebraic model for $\mathcal{T}$ (a tensor Frobenius pair), which has to satisfy a $\mathrm{K}$-theoretic regularity condition analogous to the Gersten conjecture from algebraic geometry. In this situation, we are able to prove an analogue of the Bloch formula and use it to define an intersection product similar to a construction by Grayson. We then recover the usual intersection product on a non-singular algebraic variety assuming a $\mathrm{K}$-theoretic compatibility condition.