The Stacks project

Proof. The proof given here is an example of applying algebraization and approximation. We proceed in a number of stages.

Essential surjectivity when $A$ is a complete local Noetherian ring. Let $X_ n = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/\mathfrak m^{n + 1})$. By Étale Morphisms, Theorem 41.15.2 the inclusions

induce equivalence of categories between the category of schemes étale over $X_0$ and the category of schemes étale over $X_ n$. Moreover, if $U_ n \to X_ n$ corresponds to a finite étale morphism $U_0 \to X_0$, then $U_ n \to X_ n$ is finite too, for example by More on Morphisms, Lemma 37.3.3. In this case the morphism $U_0 \to \mathop{\mathrm{Spec}}(A/\mathfrak m)$ is proper as $X_0$ is proper over $A/\mathfrak m$. Thus we may apply Grothendieck's algebraization theorem (in the form of Cohomology of Schemes, Lemma 30.28.2) to see that there is a finite morphism $U \to X$ whose restriction to $X_0$ recovers $U_0$. By More on Morphisms, Lemma 37.12.3 we see that $U \to X$ is étale at every point of $U_0$. However, since every point of $U$ specializes to a point of $U_0$ (as $U$ is proper over $A$), we conclude that $U \to X$ is étale. In this way we conclude the functor is essentially surjective.

Fully faithfulness when $A$ is a complete local Noetherian ring. Let $U \to X$ and $V \to X$ be finite étale morphisms and let $\varphi _0 : U_0 \to V_0$ be a morphism over $X_0$. Look at the morphism

This morphism is both finite étale and a closed immersion. By essential surjectivity applied to $X = U \times _ X V$ we find a finite étale morphism $W \to U \times _ X V$ whose special fibre is isomorphic to $\Gamma _{\varphi _0}$. Consider the projection $W \to U$. It is finite étale and an isomorphism over $U_0$ by construction. By Étale Morphisms, Lemma 41.14.2 $W \to U$ is an isomorphism in an open neighbourhood of $U_0$. Thus it is an isomorphism and the composition $\varphi : U \cong W \to V$ is the desired lift of $\varphi _0$.

Essential surjectivity when $A$ is a henselian local Noetherian G-ring. Let $U_0 \to X_0$ be a finite étale morphism. Let $A^\wedge $ be the completion of $A$ with respect to the maximal ideal. Let $X^\wedge $ be the base change of $X$ to $A^\wedge $. By the result above there exists a finite étale morphism $V \to X^\wedge $ whose special fibre is $U_0$. Write $A^\wedge = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A \to A_ i$ of finite type. By Limits, Lemma 32.10.1 there exists an $i$ and a finitely presented morphism $U_ i \to X_{A_ i}$ whose base change to $X^\wedge $ is $V$. After increasing $i$ we may assume that $U_ i \to X_{A_ i}$ is finite and étale (Limits, Lemmas 32.8.3 and 32.8.10). Writing

the ring map $A_ i \to A^\wedge $ can be reinterpreted as a solution $(a_1, \ldots , a_ n)$ in $A^\wedge $ for the system of equations $f_ j = 0$. By Smoothing Ring Maps, Theorem 16.13.1 we can approximate this solution (to order $11$ for example) by a solution $(b_1, \ldots , b_ n)$ in $A$. Translating back we find an $A$-algebra map $A_ i \to A$ which gives the same closed point as the original map $A_ i \to A^\wedge $ (as $11 > 1$). The base change $U \to X$ of $V \to X_{A_ i}$ by this ring map will therefore be a finite étale morphism whose special fibre is isomorphic to $U_0$.

Fully faithfulness when $A$ is a henselian local Noetherian G-ring. This can be deduced from essential surjectivity in exactly the same manner as was done in the case that $A$ is complete Noetherian.

General case. Let $(A, \mathfrak m)$ be a henselian local ring. Set $S = \mathop{\mathrm{Spec}}(A)$ and denote $s \in S$ the closed point. By Limits, Lemma 32.13.3 we can write $X \to \mathop{\mathrm{Spec}}(A)$ as a cofiltered limit of proper morphisms $X_ i \to S_ i$ with $S_ i$ of finite type over $\mathbf{Z}$. For each $i$ let $s_ i \in S_ i$ be the image of $s$. Since $S = \mathop{\mathrm{lim}}\nolimits S_ i$ and $A = \mathcal{O}_{S, s}$ we have $A = \mathop{\mathrm{colim}}\nolimits \mathcal{O}_{S_ i, s_ i}$. The ring $A_ i = \mathcal{O}_{S_ i, s_ i}$ is a Noetherian local G-ring (More on Algebra, Proposition 15.50.12). By More on Algebra, Lemma 15.12.5 we see that $A = \mathop{\mathrm{colim}}\nolimits A_ i^ h$. By More on Algebra, Lemma 15.50.8 the rings $A_ i^ h$ are G-rings. Thus we see that $A = \mathop{\mathrm{colim}}\nolimits A_ i^ h$ and

as schemes. The category of schemes finite étale over $X$ is the limit of the category of schemes finite étale over $X_ i \times _{S_ i} \mathop{\mathrm{Spec}}(A_ i^ h)$ (by Limits, Lemmas 32.10.1, 32.8.3, and 32.8.10) The same thing is true for schemes finite étale over $X_0 = \mathop{\mathrm{lim}}\nolimits (X_ i \times _{S_ i} s_ i)$. Thus we formally deduce the result for $X / \mathop{\mathrm{Spec}}(A)$ from the result for the $(X_ i \times _{S_ i} \mathop{\mathrm{Spec}}(A_ i^ h)) / \mathop{\mathrm{Spec}}(A_ i^ h)$ which we dealt with above. $\square$

Proof. The proof of this lemma is exactly the same as the proof of Lemma 58.9.1.

Essential surjectivity when $A$ is Noetherian and $I$-adically complete. Let $X_ n = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I^{n + 1})$. By Étale Morphisms, Theorem 41.15.2 the inclusions

induce equivalence of categories between the category of schemes étale over $X_0$ and the category of schemes étale over $X_ n$. Moreover, if $U_ n \to X_ n$ corresponds to a finite étale morphism $U_0 \to X_0$, then $U_ n \to X_ n$ is finite too, for example by More on Morphisms, Lemma 37.3.3. In this case the morphism $U_0 \to \mathop{\mathrm{Spec}}(A/I)$ is proper as $X_0$ is proper over $A/I$. Thus we may apply Grothendieck's algebraization theorem (in the form of Cohomology of Schemes, Lemma 30.28.2) to see that there is a finite morphism $U \to X$ whose restriction to $X_0$ recovers $U_0$. By More on Morphisms, Lemma 37.12.3 we see that $U \to X$ is étale at every point of $U_0$. However, since every point of $U$ specializes to a point of $U_0$ (as $U$ is proper over $A$), we conclude that $U \to X$ is étale. In this way we conclude the functor is essentially surjective.

Fully faithfulness when $A$ is Noetherian and $I$-adically complete. Let $U \to X$ and $V \to X$ be finite étale morphisms and let $\varphi _0 : U_0 \to V_0$ be a morphism over $X_0$. Look at the morphism

This morphism is both finite étale and a closed immersion. By essential surjectivity applied to $X = U \times _ X V$ we find a finite étale morphism $W \to U \times _ X V$ whose special fibre is isomorphic to $\Gamma _{\varphi _0}$. Consider the projection $W \to U$. It is finite étale and an isomorphism over $U_0$ by construction. By Étale Morphisms, Lemma 41.14.2 $W \to U$ is an isomorphism in an open neighbourhood of $U_0$. Thus it is an isomorphism and the composition $\varphi : U \cong W \to V$ is the desired lift of $\varphi _0$.

Essential surjectivity when $(A, I)$ is a henselian pair and $A$ is a Noetherian G-ring. Let $U_0 \to X_0$ be a finite étale morphism. Let $A^\wedge $ be the completion of $A$ with respect to $I$. Observe that $A^\wedge $ is a Noetherian ring which is $IA^\wedge $-adically complete, see Algebra, Lemmas 10.97.4 and 10.97.6. Let $X^\wedge $ be the base change of $X$ to $A^\wedge $. By the result above there exists a finite étale morphism $V \to X^\wedge $ whose special fibre is $U_0$. Write $A^\wedge = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A \to A_ i$ of finite type. By Limits, Lemma 32.10.1 there exists an $i$ and a finitely presented morphism $U_ i \to X_{A_ i}$ whose base change to $X^\wedge $ is $V$. After increasing $i$ we may assume that $U_ i \to X_{A_ i}$ is finite and étale (Limits, Lemmas 32.8.3 and 32.8.10). Writing

the ring map $A_ i \to A^\wedge $ can be reinterpreted as a solution $(a_1, \ldots , a_ n)$ in $A^\wedge $ for the system of equations $f_ j = 0$. By Smoothing Ring Maps, Lemma 16.14.1 we can approximate this solution (to order $11$ for example) by a solution $(b_1, \ldots , b_ n)$ in $A$. Translating back we find an $A$-algebra map $A_ i \to A$ which gives the same closed point as the original map $A_ i \to A^\wedge $ (as $11 > 1$). The base change $U \to X$ of $V \to X_{A_ i}$ by this ring map will therefore be a finite étale morphism whose special fibre is isomorphic to $U_0$.

Fully faithfulness when $(A, I$ is a henselian pair and $A$ is a Noetherian G-ring. This can be deduced from essential surjectivity in exactly the same manner as was done in the case that $A$ is complete Noetherian.

General case. Let $(A, I)$ be a henselian pair. Set $S = \mathop{\mathrm{Spec}}(A)$ and denote $S_0 = \mathop{\mathrm{Spec}}(A/I)$. By Limits, Lemma 32.13.3 we can write $X \to \mathop{\mathrm{Spec}}(A)$ as a cofiltered limit of proper morphisms $X_ i \to S_ i$ with $S_ i$ affine and of finite type over $\mathbf{Z}$. Write $S_ i = \mathop{\mathrm{Spec}}(A_ i)$ and denote $I_ i \subset A_ i$ the inverse image of $I$ by the map $A_ i \to A$. Set $S_{i, 0} = \mathop{\mathrm{Spec}}(A_ i/I_ i)$. Since $S = \mathop{\mathrm{lim}}\nolimits S_ i$ we have $A = \mathop{\mathrm{colim}}\nolimits A_ i$. Thus we also have $I = \mathop{\mathrm{colim}}\nolimits I_ i$ and $A/I = \mathop{\mathrm{colim}}\nolimits A_ i/I_ i$. The ring $A_ i$ is a Noetherian G-ring (More on Algebra, Proposition 15.50.12). Denote $(A_ i^ h, I_ i^ h)$ the henselization of the pair $(A_ i, I_ i)$. By More on Algebra, Lemma 15.12.5 we see that $A = \mathop{\mathrm{colim}}\nolimits A_ i^ h$. By More on Algebra, Lemma 15.50.15 the rings $A_ i^ h$ are G-rings. Thus we see that $A = \mathop{\mathrm{colim}}\nolimits A_ i^ h$ and

as schemes. The category of schemes finite étale over $X$ is the limit of the category of schemes finite étale over $X_ i \times _{S_ i} \mathop{\mathrm{Spec}}(A_ i^ h)$ (by Limits, Lemmas 32.10.1, 32.8.3, and 32.8.10) The same thing is true for schemes finite étale over $X_0 = \mathop{\mathrm{lim}}\nolimits (X_ i \times _{S_ i} S_{i, 0})$. Thus we formally deduce the result for $X / \mathop{\mathrm{Spec}}(A)$ from the result for the $(X_ i \times _{S_ i} \mathop{\mathrm{Spec}}(A_ i^ h)) / \mathop{\mathrm{Spec}}(A_ i^ h)$ which we dealt with above. $\square$

Proof. Let us prove the functor is essentially surjective. Let $U' \to X_{k'}$ be a finite étale morphism. Write $k' = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of finite type $k$-algebras. By Limits, Lemma 32.10.1 there exists an $i$ and a finitely presented morphism $U_ i \to X_{A_ i}$ whose base change to $X_{k'}$ is $U'$. After increasing $i$ we may assume that $U_ i \to X_{A_ i}$ is finite and étale (Limits, Lemmas 32.8.3 and 32.8.10). Since $k$ is algebraically closed we can find a $k$-valued point $t$ in $\mathop{\mathrm{Spec}}(A_ i)$. Let $U = (U_ i)_ t$ be the fibre of $U_ i$ over $t$. Let $A_ i^ h$ be the henselization of $(A_ i)_{\mathfrak m}$ where $\mathfrak m$ is the maximal ideal corresponding to the point $t$. By Lemma 58.9.1 we see that $(U_ i)_{A_ i^ h} = U \times \mathop{\mathrm{Spec}}(A_ i^ h)$ as schemes over $X_{A_ i^ h}$. Now since $A_ i^ h$ is algebraic over $A_ i$ (see for example discussion in Smoothing Ring Maps, Example 16.13.3) and since $k'$ is algebraically closed we can find a ring map $A_ i^ h \to k'$ extending the given inclusion $A_ i \subset k'$. Hence we conclude that $U'$ is isomorphic to the base change of $U$. The proof of fully faithfulness is exactly the same. $\square$