The following question appears, more or less, [here][1]:

Let $k$ be an algebraically closed field of characteristic zero and let $S$ be a commutative $k$-algebra
(I do not mind to further assume that $S$ is an integral domain).

Let $R_1,R_2 \subseteq S$ be two (probably different) $k$-subalgebras of $S$ such that
$R_i \subseteq S$, $1 \leq i \leq 2$ is:
finitely generated, flat and separable (in other words, $R_i \subseteq S$ is etale).

Denote the $k$-algebra generated by $R_1$ and $R_2$ by $R$.

> Is it true that $R \subseteq S$ is flat?

Clearly, $R \subseteq S$ is finitely generated and separable (hence unramified). But what about flatness?
I guess (?) that there exists a counterexample in dimension two.

A non-counterexample: $R_1=k[x^2], R_2=k[x^3], S=k[x]$; here $R=k[x^2,x^3] \subseteq k[x]$ is not flat,
but also $R_i \subseteq S$ are not separable (althouh they are finitely generated and free). See [this][2] question, which explains that $k[x^2,x^3] \subseteq k[x]$ is not separable, hence $k[x^2] \subseteq k[x]$ and $k[x^3] \subseteq k[x]$ are not separable, since if at least one of them were, then $k[x^2,x^3] \subseteq k[x]$ was separable, by a result concerning separability).

Any hints and comments are welcome; thank you very much!


  [1]: https://math.stackexchange.com/questions/3955627/flatness-of-a-subring-generated-by-two-subrings
  [2]: https://mathoverflow.net/questions/371948/separability-of-mathbbcx-over-its-mathbbc-subalgebras