The Stacks project

Proof. By Theorem 41.15.1 we see that this functor is fully faithful. It remains to show that the functor is essentially surjective. Let $Y \to S_0$ be an étale morphism of schemes.

Suppose that the result holds if $S$ and $Y$ are affine. In that case, we choose an affine open covering $Y = \bigcup V_ j$ such that each $V_ j$ maps into an affine open of $S$. By assumption (affine case) we can find étale morphisms $W_ j \to S$ such that $W_{j, 0} \cong V_ j$ (as schemes over $S_0$). Let $W_{j, j'} \subset W_ j$ be the open subscheme whose underlying topological space corresponds to $V_ j \cap V_{j'}$. Because we have isomorphisms

as schemes over $S_0$ we see by fully faithfulness that we obtain isomorphisms $\theta _{j, j'} : W_{j, j'} \to W_{j', j}$ of schemes over $S$. 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 $X \to S$ by glueing the schemes $W_ j$ along the identifications $\theta _{j, j'}$. It is clear that $X \to S$ is étale and $X_0 \cong Y$ by construction.

Thus it suffices to show the lemma in case $S$ and $Y$ are affine. Say $S = \mathop{\mathrm{Spec}}(R)$ and $S_0 = \mathop{\mathrm{Spec}}(R/I)$ with $I$ locally nilpotent. By Algebra, Lemma 10.143.2 we know that $Y$ is the spectrum of a ring $\overline{A}$ with

such that

maps to an invertible element in $\overline{A}$. Choose any lifts $f_ i \in R[x_1, \ldots , x_ n]$. Set

Since $I$ is locally nilpotent the ideal $IA$ is locally nilpotent (Algebra, Lemma 10.32.3). Observe that $\overline{A} = A/IA$. It follows that the determinant of the matrix of partials of the $f_ i$ is invertible in the algebra $A$ by Algebra, Lemma 10.32.4. Hence $R \to A$ is étale and the proof is complete. $\square$