**Proof.**
Part (1) follows from Lemma 15.93.8 but of course it follows from part (3) as well. Part (3) follows from Lemma 15.93.16 and the fact that étale $K$-algebras are finite products of finite separable extensions of $K$, see Algebra, Lemma 10.141.4. Part (3) implies (2). Part (4) follows from (3) as a product of fields is a field if and only if it has no nontrivial idempotents.

If $S \subset B$ is a subalgebra, then it is the filtered colimit of its finitely generated subalgebras which are all étale over $K$ by the above and hence $S$ is weakly étale over $K$ by Lemma 15.93.16. If $B \to Q$ is a quotient algebra, then $Q$ is the filtered colimit of $K$-algebra quotients of finite products $\prod _{i \in I} L_ i$ of finite separable extensions $L_ i/K$. Such a quotient is of the form $\prod _{i \in J} L_ i$ for some subset $J \subset I$ and hence the result holds for quotients by the same reasoning.

The statement on tensor products follows in a similar manner or by combining Lemmas 15.93.7 and 15.93.9.
$\square$

## Comments (0)