The Stacks project

Lemma 58.17.2. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Assume the completion functor

\[ \textit{Coh}(\mathcal{O}_ X) \longrightarrow \textit{Coh}(X, \mathcal{I}),\quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

is an equivalence on full subcategories of finite locally free objects (see above). Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is an equivalence.

Proof. The restriction functor is fully faithful by Lemma 58.17.1.

Let $U_1 \to Y$ be a finite étale morphism. To finish the proof we will show that $U_1$ is in the essential image of the restriction functor.

For $n \geq 1$ let $Y_ n$ be the $n$th infinitesimal neighbourhood of $Y$. By Lemma 58.8.3 there is a unique finite étale morphism $\pi _ n : U_ n \to Y_ n$ whose base change to $Y = Y_1$ recovers $U_1 \to Y_1$. Consider the sheaves $\mathcal{F}_ n = \pi _{n, *}\mathcal{O}_{U_ n}$. We may and do view $\mathcal{F}_ n$ as an $\mathcal{O}_ X$-module on $X$ wich is locally isomorphic to $(\mathcal{O}_ X/f^{n + 1}\mathcal{O}_ X)^{\oplus r}$. This $(\mathcal{F}_ n)$ is a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. By assumption there exists a finite locally free $\mathcal{O}_ X$-module $\mathcal{F}$ and a compatible system of isomorphisms

\[ \mathcal{F}/\mathcal{I}^ n\mathcal{F} \to \mathcal{F}_ n \]

of $\mathcal{O}_ X$-modules.

To construct an algebra structure on $\mathcal{F}$ consider the multiplication maps $\mathcal{F}_ n \otimes _{\mathcal{O}_ X} \mathcal{F}_ n \to \mathcal{F}_ n$ coming from the fact that $\mathcal{F}_ n = \pi _{n, *}\mathcal{O}_{U_ n}$ are sheaves of algebras. These define a map

\[ (\mathcal{F}\otimes _{\mathcal{O}_ X} \mathcal{F})^\wedge \longrightarrow \mathcal{F}^\wedge \]

in the category $\textit{Coh}(X, \mathcal{I})$. Hence by assumption we may assume there is a map $\mu : \mathcal{F}\otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{F}$ whose restriction to $Y_ n$ gives the multiplication maps above. After possibly shrinking further we may assume $\mu $ defines a commutative $\mathcal{O}_ X$-algebra structure on $\mathcal{F}$ compatible with the given algebra structures on $\mathcal{F}_ n$. Setting

\[ U = \underline{\mathop{\mathrm{Spec}}}_ X((\mathcal{F}, \mu )) \]

we obtain a finite locally free scheme $\pi : U \to X$ whose restriction to $Y$ is isomorphic to $U_1$. The the discriminant of $\pi $ is the zero set of the section

\[ \det (Q_\pi ) : \mathcal{O}_ X \longrightarrow \wedge ^{top}(\pi _*\mathcal{O}_ U)^{\otimes -2} \]

constructed in Discriminants, Section 49.3. Since the restriction of this to $Y_ n$ is an isomorphism for all $n$ by Discriminants, Lemma 49.3.1 we conclude that it is an isomorphism. Thus $\pi $ is étale by Discriminants, Lemma 49.3.1. $\square$


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