On a question of Rickard on tensor product of stably equivalent algebras

Abstract

Let $\overline\F\_p$ be the algebraic closure of the prime field of characteristic $p$. After observing that the principal block $B$ of $\overline\F\_pPSU(3,p^r)$ is stably equivalent of Morita type to its Brauer correspondent $b$, we show however that the centre of $B$ is not isomorphic as an algebra to the centre of $b$ in the cases $p^r\in \{3,4,5,8\}$. As a consequence, the algebra $B\otimes\_{\overline{\F}\_p}\overline \F\_p[X]/X^p$ is not stably equivalent of Morita type to $b\otimes\_{\overline\F\_p}\overline\F\_p[X]/X^p$ in these cases. This yields a negative answer to a question of Rickard.