Proposition 7

Let k be a field, A and B two k-algebras one of which is essentially of finite type. Suppose that A is absolutely regular (resp. absolutely normal). If the ring B is regular (resp. normal), the same is true of the ring $A \otimes_k B$.

Let K be an extension of finite type of k; let us prove that the ring $A_{(K)}$ is regular (resp. normal). Consider indeed extensions L and $k'$ having the properties of lemma 3. Then the ring $A_{(k')}$ is regular (resp. normal) by definition, and the ring $A_{(L)}$, which identifies with $A_{(k')} \otimes_{k'} L$, is regular (resp. normal) by Prop. 5, b) of No. 3. Consequently, $A_{(K)}$ is regular (resp. normal) by Prop. 5, a).

Suppose the ring B regular (resp. normal), and prove the proposition. The canonical homomorphism $B \to A \otimes_k B$ makes $A \otimes_k B$ into a free B-module. For every prime ideal $p$ of B, the ring $(A \otimes_k B) \otimes_B \kappa(p)$ identifies with $A \otimes_k \kappa(p)$; by the corollary of Prop. 9 of § 4, No. 5 (resp. Cor. 3 of Th. 4 of § 1, No. 10), it is enough to prove that $A \otimes_k \kappa(p)$ is regular (resp. normal) for every prime ideal $p$ of B.

If the $k$-algebra B is essentially of finite type, the extension $\kappa(p)$ of $k$ is of finite type and the ring $A \otimes_k \kappa(p)$ is regular (resp. normal) by what we have seen above. Suppose now the $k$-algebra A essentially of finite type; the ring $A \otimes_k \kappa(p)$ is noetherian and is the union of the increasing filtered family of noetherian subrings $A \otimes_k K$, where $K$ runs through the subextensions of finite type of $\kappa(p)$. The latter are regular (resp. normal), and one applies lemma 1 of No. 3.