The Stacks project

[IV, Theorem 18.1.2, EGA]

Theorem 41.15.2 (Une equivalence remarquable de catégories). Let $S$ be a scheme. Let $S_0 \subset S$ be a closed subscheme with the same underlying topological space (for example if the ideal sheaf of $S_0$ in $S$ has square zero). The functor

\[ X \longmapsto X_0 = S_0 \times _ S X \]

defines an equivalence of categories

\[ \{ \text{schemes }X\text{ étale over }S \} \leftrightarrow \{ \text{schemes }X_0\text{ étale over }S_0 \} \]

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

\[ W_{j, j', 0} \cong V_ j \cap V_{j'} \cong W_{j', j, 0} \]

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 10.142.2 we know that $Y$ is the spectrum of a ring $\overline{A}$ with

\[ \overline{A} = (R/I)[x_1, \ldots , x_ n]/(\overline{f}_1, \ldots , \overline{f}_ n) \]

such that

\[ \overline{g} = \det \left( \begin{matrix} \partial \overline{f}_1/\partial x_1 & \partial \overline{f}_2/\partial x_1 & \ldots & \partial \overline{f}_ n/\partial x_1 \\ \partial \overline{f}_1/\partial x_2 & \partial \overline{f}_2/\partial x_2 & \ldots & \partial \overline{f}_ n/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial \overline{f}_1/\partial x_ n & \partial \overline{f}_2/\partial x_ n & \ldots & \partial \overline{f}_ n/\partial x_ n \end{matrix} \right) \]

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

\[ A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n) \]

Since $I$ is locally nilpotent the ideal $IA$ is locally nilpotent (Algebra, Lemma 10.31.3). 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 10.31.4. Hence $R \to A$ is étale and the proof is complete. $\square$


Comments (2)

Comment #2726 by on

A reference is EGA IV_45, Theorem 18.1.2


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