Rewriting a tensor product as a quotient of a polynomial ring Good evening.
While trying to show that the base change of étale morphisms is étale, I have ended up stuck on a commutative algebra question I can't seem to be able to solve.
Suppose you are given two $S$-algebras, $S'$ and $R'$, and that you know that $$ R' = R[x_1,\dots, x_n]/(f_1, ... f_n) $$ for some other $S$-algebra $R$.
Can you express the algebra $R' \otimes_{S} S'$ as a quotient of a polynomial ring with coefficients in $R \otimes_S S'$?

Yes. We have
$R[x_1,\dotsc,x_n]/\langle f_1,\dotsc,f_n \rangle \otimes_S S' = (R[x_1,\dotsc,x_n] \otimes_S S')/ \langle f_1 \otimes 1,\dotsc,f_n \otimes 1 \rangle$
and
$~~~~R[x_1,\dotsc,x_n] \otimes_S S' \\
= (S[x_1,\dotsc,x_n] \otimes_S R) \otimes_S S'\\ = S[x_1,\dotsc,x_n] \otimes_S (R \otimes_S S')\\ = (R \otimes_S S')[x_1,\dotsc,x_n].$
