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 10.143.2 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 10.32.3). 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. $\square$

Comments (2)

Comment #2726 by on

A reference is EGA IV_45, Theorem 18.1.2

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 039R. Beware of the difference between the letter 'O' and the digit '0'.