Let $K$ be a field, $\alpha, \beta \in \mathrm{Br}(K)$, let $X,Y$ be their Brauer-Severi Varieties, is there a way to calculate $A^\*(X\times Y)$? 


For example, if $\alpha,\beta$ both has degree $5$, $2\alpha=\beta$, then $A^\*(X\times Y)$ is a subring (Will two non-rational equivalents cycles become rational equivalent after base change?) of $A^\*(X\times Y\_{\overline{K}})=A^\*(\mathbb{P^4}\times\mathbb{P^4})=Z[H\_1,H\_2]/(H\_1^5,H\_2^5)$. Then $5H\_i\in A^1$, $H\_i\notin A^1$, but $4H\_1+3H\_2\in A^1$, as it is the first Chern class of the vector bundle bundle $O(1)\boxtimes (\Omega\otimes O(2))$ on $X\times Y$. 



Chow ring of product of Brauer-Severi Varieties
In general I think it is fairly subtle.


There are some partial results though.
If $\beta$ is in the subgroup of $\text{Br}(K)$ generated by $\alpha$ then the ring $\text{CH}(X\times Y)$ is isomorphic to a direct sum of shifted copies of $\text{CH}(X)$. In this case, the claim is due to the fact the product of Severi-Brauer varieties $X\times Y$ is isomorphic to a projective bundle over $X$.


Let me assume $\alpha\neq 0$. Let $A$ be the algebra representing $\alpha$ and $B$ be the algebra representing $\beta$. A complete calculation of the ring in this case can be obtained as follows: for a prime degree algebra $A$ the Chow ring is generated by chern classes of the tautological vector bundle of rank the degree of this algebra. So, denoting $X=\text{SB}(A)$, a description $\text{CH}(X)=\mathbf{Z}\oplus \mathbf{Z}e\oplus \mathbf{Z}c\_2\oplus \mathbf{Z}c\_3\oplus \mathbf{Z}c\_4$ can be obtained where under the pullback to an algebraic closure, say $\text{CH}(\mathbf{P}^4)$, the elements $e,c\_2,c\_3,c\_4$ map to $5h,5h^2,5h^3,5h^4$ respectively. By the projective bundle formula $\text{CH}(X\times Y)$ is isomorphic with $\text{CH}(X)[\xi]/(\xi^5+ s\_1\xi^4 + s\_2\xi^3 + s\_3\xi^2 + s\_4\xi + s\_5)$ where $s\_i$ are the $i$th Chern classes of a bundle realizing this product as a projective bundle over $X$. It just remains to find the $s\_i$.


Let $\mathcal{T}\_A$ be the tautological bundle of rank the degree of $A$ on $Y=\text{SB}(A)$. It's square $\mathcal{T}\_A^{\otimes 2}$ is rank 25 bundle that splits into the sum of 5 indecomposable (and isomorphic) rank 5 bundles, which I'll call $\eta\_2$. The projective bundle $X\times Y\rightarrow X$ is isomorphic with $\mathbf{P}(\eta\_2)\rightarrow X$ and the Chern classes are determined as $s\_i=c\_i(\eta\_2)$. The pullback of $c\_i(\eta\_2)$ to $\text{CH}(\mathbf{P}^4)$ is mapped to $c\_i(\mathcal{O}(2)^{\oplus 5})= \binom{5}{i}2^i h^i$ so that $s\_1= 2e$, $s\_2= 8c\_2$, $s\_3= 16c\_3$, $s\_4= 16c\_4$, and $s\_5=0$.


