Let $D$ be a central division algebra over $\mathbb{Q}_p$, of dimension $n^2$. For example, let $D=E_n(\Pi)$, where $E_n$ is the unramified degree extension of $\mathbb{Q}_p$ such that $\Pi^n=p$ and $\Pi(x)=\varphi(x)\Pi$. Let $R$ be any commutative $\mathbb{Q}_p$-algebra. Assume there is only one isomorphic class of simple $D\otimes_{\mathbb{Q}_p}R$-module such that it is a free $R$-module of rank $n$. Is $D\otimes_{\mathbb{Q}_p}R$ necessarily split, i.e $D\otimes_{\mathbb{Q}_p}R\simeq M_n(R)$? We know $R^n$ has only one possible $M_n(R)$-module structure up to isomorphic, but is the converse true?
0
Add a comment
|