The Stacks project

Proof. It is immediate that (2) implies (1). Assume $\{ Y_ i \to Y\} $ is as in (1) and let $W \to Y$ be as in (2). Then $\{ Y_ i \times _ Y W \to W\} _{i \in I}$ is an étale covering, which we may refine by an étale covering $\{ W_ j \to W\} _{j = 1, \ldots , m}$ with $W_ j$ affine (Topologies, Lemma 34.4.4). Thus to finish the proof it suffices to show the following three algebraic statements:

1. if $R \to A \to B$ are ring maps with $A \to B$ étale and $A$ glueable as an $R$-module, then $B$ is glueable as an $R$-module,

2. finite products of glueable $R$-modules are glueable,

3. if $R \to A \to B$ are ring maps with $A \to B$ faithfully étale and $B$ glueable as an $R$-module, then $A$ is glueable as an $R$-module.

Namely, the first of these will imply that $\Gamma (W_ j, \mathcal{O}_{W_ j})$ is a glueable $R$-module, the second will imply that $\prod \Gamma (W_ j, \mathcal{O}_{W_ j})$ is a glueable $R$-module, and the third will imply that $\Gamma (W, \mathcal{O}_ W)$ is a glueable $R$-module.

Consider an étale $R$-algebra homomorphism $A \to B$. Set $A' = A \otimes _ R R'$ and $B' = B \otimes _ R R' = A' \otimes _ A B$. Statements (1) and (3) then follow from the following facts: (a) $A$, resp. $B$ is glueable if and only if the sequence

is exact, (b) the second sequence is equal to the functor $- \otimes _ A B$ applied to the first and (c) (faithful) flatness of $A \to B$. We omit the proof of (2). $\square$