Corollary 3

Let $k$ be a field, $A$ a $k$-algebra, and $K$ an extension of $k$. Suppose that the $k$-algebra $A$ is essentially of finite type or that the extension $K$ of $k$ is of finite type. In order that the $k$-algebra $A$ be absolutely regular (resp. absolutely normal), it is necessary and sufficient that the $K$-algebra $A_{(K)}$ be so.

Suppose $A$ absolutely regular (resp. absolutely normal) and let $K'$ be a purely inseparable extension of $K$ of finite degree. The ring $K' \otimes_K A_{(K)}$, isomorphic to $K' \otimes_k A$, is regular (resp. normal) by cor. 2.

Suppose conversely that the $K$-algebra $A_{(K)}$ is absolutely regular (resp. absolutely normal), and let $k'$ be a purely inseparable extension of $k$ of finite degree. Let $L$ be a composite extension of $k'$ and $K$; then the ring $A_{(L)}$ may be identified with $L \otimes_K A_{(K)}$, hence is regular (resp. normal); consequently, the ring $A_{(k')}$ is regular (resp. normal) by prop. 5, a) of No. 3.