The Stacks project

Lemma 58.17.4. Let $X$ be a Noetherian scheme and let $Y \subset X$ be a closed subscheme with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{V}$ be the set of open subschemes $V \subset X$ containing $Y$ ordered by reverse inclusion. Assume the completion functor

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

defines an equivalence of the full subcategories of finite locally free objects (see explanation above). Then the restriction functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{FÉt}_ V \to \textit{FÉt}_ Y \]

is an equivalence.

Proof. Observe that $\mathcal{V}$ is a directed set, so the colimits are as in Categories, Section 4.19. The rest of the argument is almost exactly the same as the argument in the proof of Lemma 58.17.2; we urge the reader to skip it.

The restriction functor is fully faithful by Lemma 58.17.3.

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 $X$ which 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 $V \in \mathcal{V}$ and a finite locally free $\mathcal{O}_ V$-module $\mathcal{F}$ and a compatible system of isomorphisms

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

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

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

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

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

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

we obtain a finite locally free scheme over $V$ whose restriction to $Y$ is isomorphic to $U_1$. It follows that $U \to V$ is étale at all points lying over $Y$, see More on Morphisms, Lemma 37.12.3. Thus after shrinking $V$ once more we may assume $U \to V$ is finite étale. This finishes the proof. $\square$

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