The Stacks project

Proof. Let $P'$ be one of the following properties of homomorphisms of rings: “finite”, “surjective”, “flat”, “finite type”, “flat and of finite presentation”, “étale”. Translated into algebra, the statement means the following: If $A \to B$ is an $R$-algebra homomorphism and $A$ and $B$ are glueable for $(R \to R', f)$, then $A_ f \to B_ f$ and $A \otimes _ R R' \to B \otimes _ R R'$ have $P'$ if and only if $A \to B$ has $P'$.

By More on Algebra, Lemmas 15.90.4 and 15.90.18 the algebraic statement is true for $P'$ equal to “finite” or “flat”.

If $A_ f \to B_ f$ and $A \otimes _ R R' \to B \otimes _ R R'$ are surjective, then $N = B/A$ is an $R$-module with $N_ f = 0$ and $N \otimes _ R R' = 0$ and hence vanishes by More on Algebra, Lemma 15.90.2. Thus $A \to B$ is surjective.

If $A_ f \to B_ f$ and $A \otimes _ R R' \to B \otimes _ R R'$ are finite type, then we can choose an $A$-algebra homomorphism $A[x_1, \ldots , x_ n] \to B$ such that $A_ f[x_1, \ldots , x_ n] \to B_ f$ and $(A \otimes _ R R')[x_1, \ldots , x_ n] \to B \otimes _ R R'$ are surjective (small detail omitted). We conclude that $A[x_1, \ldots , x_ n] \to B$ is surjective by the previous result. Thus $A \to B$ is of finite type.

If $A_ f \to B_ f$ and $A \otimes _ R R' \to B \otimes _ R R'$ are flat and of finite presentation, then we know that $A \to B$ is flat and of finite type by what we have already shown. Choose a surjection $A[x_1, \ldots , x_ n] \to B$ and denote $I$ the kernel. By flatness of $B$ over $A$ we see that $I_ f$ is the kernel of $A_ f[x_1, \ldots , x_ n] \to B_ f$ and $I \otimes _ R R'$ is the kernel of $A \otimes _ R R'[x_1, \ldots , x_ n] \to B \otimes _ R R'$. Thus $I_ f$ is a finite $A_ f[x_1, \ldots , x_ n]$-module and $I \otimes _ R R'$ is a finite $(A \otimes _ R R')[x_1, \ldots , x_ n]$-module. By More on Algebra, Lemma 15.90.4 applied to $I$ viewed as a module over $A[x_1, \ldots , x_ n]$ we conclude that $I$ is a finitely generated ideal and we conclude $A \to B$ is flat and of finite presentation.

If $A_ f \to B_ f$ and $A \otimes _ R R' \to B \otimes _ R R'$ are étale, then we know that $A \to B$ is flat and of finite presentation by what we have already shown. Since the fibres of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ are isomorphic to fibres of $\mathop{\mathrm{Spec}}(B_ f) \to \mathop{\mathrm{Spec}}(A_ f)$ or $\mathop{\mathrm{Spec}}(B/fB) \to \mathop{\mathrm{Spec}}(A/fA)$, we conclude that $A \to B$ is unramified, see Morphisms, Lemmas 29.35.11 and 29.35.12. We conclude that $A \to B$ is étale by Morphisms, Lemma 29.36.16 for example. $\square$