The Stacks project

First proof. By Theorem 59.45.1 we see that the functor is fully faithful. It remains to show that the functor is essentially surjective. Let $U \to X$ be an étale morphism of schemes.

Suppose that the result holds if $U$ and $Y$ are affine. In that case, we choose an affine open covering $U = \bigcup U_ i$ such that each $U_ i$ maps into an affine open of $Y$. By assumption (affine case) we can find étale morphisms $V_ i \to Y$ such that $X \times _ Y V_ i \cong U_ i$ as schemes over $X$. Let $V_{i, i'} \subset V_ i$ be the open subscheme whose underlying topological space corresponds to $U_ i \cap U_{i'}$. Because we have isomorphisms

as schemes over $X$ we see by fully faithfulness that we obtain isomorphisms $\theta _{i, i'} : V_{i, i'} \to V_{i', i}$ of schemes over $Y$. We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section 26.14. Applying Schemes, Lemma 26.14.2 we obtain a scheme $V \to Y$ by glueing the schemes $V_ i$ along the identifications $\theta _{i, i'}$. It is clear that $V \to Y$ is étale and $X \times _ Y V \cong U$ by construction.

Thus it suffices to show the lemma in case $U$ and $Y$ are affine. Recall that in the proof of Theorem 59.45.1 we showed that $U$ comes with a unique descent datum $(U, \varphi )$ relative to $X/Y$. By Étale Morphisms, Proposition 41.20.6 (which applies because $U \to X$ is quasi-compact and separated as well as étale by our reduction to the affine case) there exists an étale morphism $V \to Y$ such that $X \times _ Y V \cong U$ and the proof is complete. $\square$

Second proof. By Theorem 59.45.1 we see that the functor is fully faithful. It remains to show that the functor is essentially surjective. Let $U \to X$ be an étale morphism of schemes.

Suppose that the result holds if $U$ and $Y$ are affine. In that case, we choose an affine open covering $U = \bigcup U_ i$ such that each $U_ i$ maps into an affine open of $Y$. By assumption (affine case) we can find étale morphisms $V_ i \to Y$ such that $X \times _ Y V_ i \cong U_ i$ as schemes over $X$. Let $V_{i, i'} \subset V_ i$ be the open subscheme whose underlying topological space corresponds to $U_ i \cap U_{i'}$. Because we have isomorphisms

as schemes over $X$ we see by fully faithfulness that we obtain isomorphisms $\theta _{i, i'} : V_{i, i'} \to V_{i', i}$ of schemes over $Y$. We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section 26.14. Applying Schemes, Lemma 26.14.2 we obtain a scheme $V \to Y$ by glueing the schemes $V_ i$ along the identifications $\theta _{i, i'}$. It is clear that $V \to Y$ is étale and $X \times _ Y V \cong U$ by construction.

Thus it suffices to prove that the functor

is essentially surjective when $X$ and $Y$ are affine.

Let $U \to X$ be an affine scheme étale over $X$. We have to find $V \to Y$ étale (and affine) such that $X \times _ Y V$ is isomorphic to $U$ over $X$. Note that an étale morphism of affines has universally bounded fibres, see Morphisms, Lemmas 29.36.6 and 29.57.9. Hence we can do induction on the integer $n$ bounding the degree of the fibres of $U \to X$. See Morphisms, Lemma 29.57.8 for a description of this integer in the case of an étale morphism. If $n = 1$, then $U \to X$ is an open immersion (see Étale Morphisms, Theorem 41.14.1), and the result is clear. Assume $n > 1$.

By Lemma 59.44.4 there exists an étale morphism of schemes $W \to Y$ and a surjective morphism $W_ X \to U$ over $X$. As $U$ is quasi-compact we may replace $W$ by a disjoint union of finitely many affine opens of $W$, hence we may assume that $W$ is affine as well. Here is a diagram

The disjoint union decomposition arises because by construction the étale morphism of affine schemes $U \times _ Y W \to W_ X$ has a section. OK, and now we see that the morphism $R \to X \times _ Y W$ is an étale morphism of affine schemes whose fibres have degree universally bounded by $n - 1$. Hence by induction assumption there exists a scheme $V' \to W$ étale such that $R \cong W_ X \times _ W V'$. Taking $V'' = W \amalg V'$ we find a scheme $V''$ étale over $W$ whose base change to $W_ X$ is isomorphic to $U \times _ Y W$ over $X \times _ Y W$.

At this point we can use descent to find $V$ over $Y$ whose base change to $X$ is isomorphic to $U$ over $X$. Namely, by the fully faithfulness of the functor (59.45.2.1) corresponding to the universal homeomorphism $X \times _ Y (W \times _ Y W) \to (W \times _ Y W)$ there exists a unique isomorphism $\varphi : V'' \times _ Y W \to W \times _ Y V''$ whose base change to $X \times _ Y (W \times _ Y W)$ is the canonical descent datum for $U \times _ Y W$ over $X \times _ Y W$. In particular $\varphi $ satisfies the cocycle condition. Hence by Descent, Lemma 35.37.1 we see that $\varphi $ is effective (recall that all schemes above are affine). Thus we obtain $V \to Y$ and an isomorphism $V'' \cong W \times _ Y V$ such that the canonical descent datum on $W \times _ Y V/W/Y$ agrees with $\varphi $. Note that $V \to Y$ is étale, by Descent, Lemma 35.23.29. Moreover, there is an isomorphism $V_ X \cong U$ which comes from descending the isomorphism

which we have by construction. Some details omitted. $\square$