Products of Brauer Severi surfaces

Abstract

Let $\{P_i{\}}_{1\le i\le r}$ and $\{Q_i{\}}_{1\le i\le r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_i^{\prime }s$ in $Br(k)$ is the same as the subgroup generated by the $Q_i^{\prime }s$ \iff $\Pi P_i$ is birational to $\Pi Q_i$. Moreover in this case $\Pi P_i$ and $\Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.

7 pages