Splitting fields and separability

Abstract:

It is a classical result that if $(R,\mathfrak {M})$ is a complete discrete valuation ring with quotient field $K$, and if $R/\mathfrak {M}$ is perfect, then any finite dimensional central simple $K$-algebra $\Sigma$ can be split by a field $L$ which is an unramified extension of $K$. Here we prove that if $(R,\mathfrak {M})$ is any regular local ring, and if $\Sigma$ contains an $R$-order $\Lambda$ whose global dimension is finite and such that $\Lambda /\operatorname {Rad} \Lambda$ is central simple over $R/\mathfrak {M}$, then the existence of an “$R$-unramified” splitting field $L$ for $\Sigma$ implies that $\Lambda$ is $R$-separable. Using this theorem we construct an example which shows that if $R$ is a regular local ring of dimension greater than one, and if its characteristic is not 2, then there is a central division algebra over $K$ which has no $R$-unramified splitting field.