The Stacks project

Lemma 58.9.1. Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$ with closed fibre $X_0$. Then the functor

\[ \textit{FÉt}_ X \to \textit{FÉt}_{X_0},\quad U \longmapsto U_0 = U \times _ X X_0 \]

is an equivalence of categories.

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

\[ X_0 \to X_1 \to X_2 \to \ldots \]

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

\[ \Gamma _{\varphi _0} : U_0 \longrightarrow U_0 \times _{X_0} V_0 \]

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

\[ A_ i = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m) \]

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

\[ X = \mathop{\mathrm{lim}}\nolimits (X_ i \times _{S_ i} \mathop{\mathrm{Spec}}(A_ i^ h)) \]

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. $\square$

Comments (3)

Comment #6764 by Laurent Moret-Bailly on

This should extend to henselian pairs, right? This would involve extending theorem 07QY (approximation) to henselian pairs where is a noetherian G-ring, and extending (part of) Lemma 07QR to show that the henselization of a noetherian G-ring along any ideal is a G-ring. Am I missing something?

Comment #6766 by on

Yes, the statement you suggest sounds correct. Fun! I guess the proof as you sketched it should work too. The proof will use Lemmas 15.50.14 and 15.50.15 and (as you say) the extension of Theorem 15.50.8 to henselian pairs, namely Lemma 16.14.1. I will add this the next time I go through all the comments. Thanks very much!

Comment #6927 by on

OK, I added this result but it doesn't have a tag yet. It will in a few days appear as the lemma following this one.

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