Corollary 2

Let $k$ be a field, $A$ an absolutely regular (resp. absolutely normal) $k$-algebra, and $K$ an extension of $k$. Suppose that $A$ is essentially of finite type or that the extension $K$ of $k$ is of finite type.

a) The ring $A_{(K)}$ is regular (resp. normal).

b) If the extension $K$ of $k$ is separable, the $k$-algebra $A_{(K)}$ is absolutely regular (resp. absolutely normal).

Assertion a) results from proposition 7; assertion b) results from cor. 1 and example 2.