Proposition 5

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

¹ In this number, an algebra over a field $k$ is said to be regular if it is a regular ring. In particular, every extension of $k$ is a regular algebra. Care will be taken not to confuse this notion with that of regular extension, introduced in A, V, p. 135, def. 2, which will not occur here.

a) If the ring $A_{(K)}$ is regular (resp. normal), the ring $A$ is regular (resp. normal).

b) If the ring $A$ is regular (resp. normal) and if the extension $K$ of $k$ is separable, the ring $A_{(K)}$ is regular (resp. normal).

The $A$-module $A_{(K)}$ is free, hence faithfully flat; assertion a) therefore follows from cor. of prop. 8 of I, § 3, No. 5 and from prop. 8 of § 4, No. 5 (resp. cor. 2 of th. 4 of § 1, No. 10). Under the hypotheses of b), for every prime ideal $\mathfrak{p}$ of $A$, the ring $\kappa(\mathfrak{p}) \otimes_k K$ is regular (lemma 2), and $a fortiori$ normal (§ 4, No. 2, cor. 2 of prop. 4). Assertion b) therefore follows from cor. of prop. 9 of § 4, No. 5 (resp. cor. 3 of th. 4 of § 1, No. 10).