The Stacks project

Proof. To prove (a), choose a finite set $\{ x_ i\} $ of generators of $A \otimes _ R S$ over $S$. Write each $x_ i$ as $\sum _ j y_{ij} \otimes s_{ij}$ with $y_{ij} \in A$ and $s_{ij} \in S$. Let $F$ be the polynomial $R$-algebra on variables $e_{ij}$ and let $F \to M$ be the $R$-algebra map sending $e_{ij}$ to $y_{ij}$. Then $F \otimes _ R S\to A \otimes _ R S$ is surjective, so $\mathop{\mathrm{Coker}}(F \to A) \otimes _ R S$ is zero and hence $\mathop{\mathrm{Coker}}(F \to A)$ is zero. This proves (a).

To see (b) assume $A \otimes _ R S$ is a finitely presented $S$-algebra. Then $A$ is finite type over $R$ by (a). Choose a surjection $R[x_1, \ldots , x_ n] \to A$ with kernel $I$. Then $I \otimes _ R S \to S[x_1, \ldots , x_ n] \to A \otimes _ R S \to 0$ is exact. By Algebra, Lemma 10.6.3 the kernel of $S[x_1, \ldots , x_ n] \to A \otimes _ R S$ is a finitely generated ideal. Thus we can find finitely many elements $y_1, \ldots , y_ t \in I$ such that the images of $y_ i \otimes 1$ in $S[x_1, \ldots , x_ n]$ generate the kernel of $S[x_1, \ldots , x_ n] \to A \otimes _ R S$. Let $I' \subset I$ be the ideal generated by $y_1, \ldots , y_ t$. Then $A' = R[x_1, \ldots , x_ n]/I'$ is a finitely presented $R$-algebra with a morphism $A' \to A$ such that $A' \otimes _ R S \to A \otimes _ R S$ is an isomorphism. Thus $A' \cong A$ as desired.

To prove (c), recall that $A$ is formally unramified over $R$ if and only if the module of relative differentials $\Omega _{A/R}$ vanishes, see Algebra, Lemma 10.148.2 or [Proposition 17.2.1, EGA4]. Since $\Omega _{(A \otimes _ R S)/S} = \Omega _{A/R} \otimes _ R S$, the vanishing descends by Theorem 35.4.22.

To deduce (d) from the previous cases, recall that $A$ is unramified over $R$ if and only if $A$ is formally unramified and of finite type over $R$, see Algebra, Lemma 10.151.2.

To prove (e), recall that by Algebra, Lemma 10.151.8 or [Théorème 17.6.1, EGA4] the algebra $A$ is étale over $R$ if and only if $A$ is flat, unramified, and of finite presentation over $R$. $\square$