The Stacks project

Proof. Since the category of finite étale coverings has an internal hom (Lemma 58.5.4) it suffices to prove the following: Given $U$ finite étale over $X$ and a morphism $t : Y \to U$ over $X$ there exists a unique section $s : X \to U$ such that $t = s|_ Y$. Picture

Finding the dotted arrow $s$ is the same thing as finding an $\mathcal{O}_ X$-algebra map

which reduces modulo the ideal sheaf of $Y$ to the given algebra map $t^\sharp : f_*\mathcal{O}_ U \to \mathcal{O}_ Y$. By Lemma 58.8.3 we can lift $t$ uniquely to a compatible system of maps $t_ n : Y_ n \to U$ and hence a map

of sheaves of algebras on $X$. Since $f_*\mathcal{O}_ U$ is a finite locally free $\mathcal{O}_ X$-module, we conclude that we get a unique $\mathcal{O}_ X$-module map $\sigma : f_*\mathcal{O}_ U \to \mathcal{O}_ X$ whose completion is $\mathop{\mathrm{lim}}\nolimits t_ n^\sharp $. To see that $\sigma $ is an algebra homomorphism, we need to check that the diagram

commutes. For every $n$ we know this diagram commutes after restricting to $Y_ n$, i.e., the diagram commutes after applying the completion functor. Hence by faithfulness of the completion functor we conclude. $\square$

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

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

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. By faithfulness of the functor in the statement of the lemma, we conclude that $\mu $ defines a commutative $\mathcal{O}_ X$-algebra structure on $\mathcal{F}$ compatible with the given algebra structures on $\mathcal{F}_ n$. Setting

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

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. $\square$

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 58.17.1; we urge the reader to skip it.

Since the category of finite étale coverings has an internal hom (Lemma 58.5.4) it suffices to prove the following: Given $U$ finite étale over $V \in \mathcal{V}$ and a morphism $t : Y \to U$ over $V$ there exists a $V' \geq V$ and a morphism $s : V' \to U$ over $V$ such that $t = s|_ Y$. Picture

Finding the dotted arrow $s$ is the same thing as finding an $\mathcal{O}_{V'}$-algebra map

which reduces modulo the ideal sheaf of $Y$ to the given algebra map $t^\sharp : f_*\mathcal{O}_ U \to \mathcal{O}_ Y$. By Lemma 58.8.3 we can lift $t$ uniquely to a compatible system of maps $t_ n : Y_ n \to U$ and hence a map

of sheaves of algebras on $V$. Observe that $f_*\mathcal{O}_ U$ is a finite locally free $\mathcal{O}_ V$-module. Hence we get a $V' \geq V$ a map $\sigma : f_*\mathcal{O}_ U|_{V'} \to \mathcal{O}_{V'}$ whose completion is $\mathop{\mathrm{lim}}\nolimits t_ n^\sharp $. To see that $\sigma $ is an algebra homomorphism, we need to check that the diagram

commutes. For every $n$ we know this diagram commutes after restricting to $Y_ n$, i.e., the diagram commutes after applying the completion functor. Hence by faithfulness of the completion functor we deduce that there exists a $V'' \geq V'$ such that $\sigma |_{V''}$ is an algebra homomorphism as desired. $\square$

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

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

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

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. $\square$

Proof. Let $a, b : U \to U'$ be two morphisms of schemes finite étale over $X$ whose restriction to $Y$ are the same. The image of a connected component of $U$ is an connected component of $X$; this follows from Topology, Lemma 5.7.7 applied to the restriction of $U \to X$ to a connected component of $X$. Hence the image of every connected component of $U$ meets $Y$ by assumption. We conclude that $a = b$ after restriction to each connected component of $U$ by Étale Morphisms, Proposition 41.6.3. Since the equalizer of $a$ and $b$ is an open subscheme of $U$ (as the diagonal of $U'$ over $X$ is open) we conclude. $\square$

Proof. This lemma follows formally from Lemma 58.17.1 and Algebraic and Formal Geometry, Lemma 52.15.1. $\square$

Proof. This lemma follows formally from Lemma 58.17.3 and Algebraic and Formal Geometry, Lemma 52.15.2. $\square$