Exercise 2, § 6

Let $k$ be a field, A an essentially of finite type $k$-algebra, B a $k$-algebra, $p$ an integer.

a) If A and B satisfy the property $S_p$ ($§ 1$, Exercise 8), the same is true of $A \otimes_k B$ (reason in the same way as in the proof of Proposition 4, using loc. cit., f)).

b) Let K be an extension of $k$; for $A_{(K)}$ to satisfy $S_p$, it is necessary and sufficient that this be so for A.