## 10.156 Henselization and quasi-finite ring maps

In this section we prove some results concerning the functorial maps between (strict) henselizations for quasi-finite ring maps.

Lemma 10.156.1. Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over $\mathfrak p$ in $R$. Assume $R \to S$ is quasi-finite at $\mathfrak q$. The commutative diagram

$\xymatrix{ R_{\mathfrak p}^ h \ar[r] & S_{\mathfrak q}^ h \\ R_{\mathfrak p} \ar[u] \ar[r] & S_{\mathfrak q} \ar[u] }$

of Lemma 10.155.6 identifies $S_{\mathfrak q}^ h$ with the localization of $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$ at the prime generated by $\mathfrak q$. Moreover, the ring map $R_{\mathfrak p}^ h \to S_{\mathfrak q}^ h$ is finite.

Proof. Note that $R_{\mathfrak p}^ h \otimes _ R S$ is quasi-finite over $R_{\mathfrak p}^ h$ at the prime ideal corresponding to $\mathfrak q$, see Lemma 10.122.6. Hence the localization $S'$ of $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$ is henselian and finite over $R_{\mathfrak p}^ h$, see Lemma 10.153.4. As a localization $S'$ is a filtered colimit of étale $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$-algebras. By Lemma 10.155.8 we see that $S_\mathfrak q^ h$ is the henselization of $R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$. Thus $S' = S_\mathfrak q^ h$ by the uniqueness result of Lemma 10.154.7. $\square$

Lemma 10.156.2. Let $R$ be a local ring with henselization $R^ h$. Let $I \subset \mathfrak m_ R$. Then $R^ h/IR^ h$ is the henselization of $R/I$.

Proof. This is a special case of Lemma 10.156.1. $\square$

Lemma 10.156.3. Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over $\mathfrak p$ in $R$. Assume $R \to S$ is quasi-finite at $\mathfrak q$. Let $\kappa _2^{sep}/\kappa (\mathfrak q)$ be a separable algebraic closure and denote $\kappa _1^{sep} \subset \kappa _2^{sep}$ the subfield of elements separable algebraic over $\kappa (\mathfrak q)$ (Fields, Lemma 9.14.6). The commutative diagram

$\xymatrix{ R_{\mathfrak p}^{sh} \ar[r] & S_{\mathfrak q}^{sh} \\ R_{\mathfrak p} \ar[u] \ar[r] & S_{\mathfrak q} \ar[u] }$

of Lemma 10.155.10 identifies $S_{\mathfrak q}^{sh}$ with the localization of $R_{\mathfrak p}^{sh} \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$ at the prime ideal which is the kernel of the map

$R_{\mathfrak p}^{sh} \otimes _{R_{\mathfrak p}} S_{\mathfrak q} \longrightarrow \kappa _1^{sep} \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak q) \longrightarrow \kappa _2^{sep}$

Moreover, the ring map $R_{\mathfrak p}^{sh} \to S_{\mathfrak q}^{sh}$ is a finite local homomorphism of local rings whose residue field extension is the extension $\kappa _2^{sep}/\kappa _1^{sep}$ which is both finite and purely inseparable.

Proof. Since $R \to S$ is quasi-finite at $\mathfrak q$ we see that the extension $\kappa (\mathfrak q)/\kappa (\mathfrak p)$ is finite, see Definition 10.122.3 and Lemma 10.122.2. Hence $\kappa _1^{sep}$ is a separable algebraic closure of $\kappa (\mathfrak p)$ (small detail omitted). In particular Lemma 10.155.10 does really apply. Next, the compositum of $\kappa (\mathfrak p)$ and $\kappa _1^{sep}$ in $\kappa _2^{sep}$ is separably algebraically closed and hence equal to $\kappa _2^{sep}$. We conclude that $\kappa _2^{sep}/\kappa _1^{sep}$ is finite. By construction the extension $\kappa _2^{sep}/\kappa _1^{sep}$ is purely inseparable. The ring map $R_{\mathfrak p}^{sh} \to S_{\mathfrak q}^{sh}$ is indeed local and induces the residue field extension $\kappa _2^{sep}/\kappa _1^{sep}$ which is indeed finite purely inseparable.

Note that $R_{\mathfrak p}^{sh} \otimes _ R S$ is quasi-finite over $R_{\mathfrak p}^{sh}$ at the prime ideal $\mathfrak q'$ given in the statement of the lemma, see Lemma 10.122.6. Hence the localization $S'$ of $R_{\mathfrak p}^{sh} \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$ at $\mathfrak q'$ is henselian and finite over $R_{\mathfrak p}^{sh}$, see Lemma 10.153.4. Note that the residue field of $S'$ is $\kappa _2^{sep}$ as the map $\kappa _1^{sep} \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak q) \to \kappa _2^{sep}$ is surjective by the discussion in the previous paragraph. Furthermore, as a localization $S'$ is a filtered colimit of étale $R_{\mathfrak p}^{sh} \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$-algebras. By Lemma 10.155.12 we see that $S_{\mathfrak q}^{sh}$ is the strict henselization of $R_{\mathfrak p}^{sh} \otimes _{R_{\mathfrak p}} S_{\mathfrak q}$ at $\mathfrak q'$. Thus $S' = S_\mathfrak q^{sh}$ by the uniqueness result of Lemma 10.154.7. $\square$

Lemma 10.156.4. Let $R$ be a local ring with strict henselization $R^{sh}$. Let $I \subset \mathfrak m_ R$. Then $R^{sh}/IR^{sh}$ is a strict henselization of $R/I$.

Proof. This is a special case of Lemma 10.156.3. $\square$

Lemma 10.156.5. Let $A \to B$ and $A \to C$ be local homomorphisms of local rings. If $A \to C$ is integral and either $\kappa (\mathfrak m_ C)/\kappa (\mathfrak m_ A)$ or $\kappa (\mathfrak m_ B)/\kappa (\mathfrak m_ A)$ is purely inseparable, then $D = B \otimes _ A C$ is a local ring and $B \to D$ and $C \to D$ are local.

Proof. Any maximal ideal of $D$ lies over the maximal ideal of $B$ by going up for the integral ring map $B \to D$ (Lemma 10.36.22). Now $D/\mathfrak m_ B D = \kappa (\mathfrak m_ B) \otimes _ A C = \kappa (\mathfrak m_ B) \otimes _{\kappa (\mathfrak m_ A)} C/\mathfrak m_ A C$. The spectrum of $C/\mathfrak m_ A C$ consists of a single point, namely $\mathfrak m_ C$. Thus the spectrum of $D/\mathfrak m_ B D$ is the same as the spectrum of $\kappa (\mathfrak m_ B) \otimes _{\kappa (\mathfrak m_ A)} \kappa (\mathfrak m_ C)$ which is a single point by our assumption that either $\kappa (\mathfrak m_ C)/\kappa (\mathfrak m_ A)$ or $\kappa (\mathfrak m_ B)/\kappa (\mathfrak m_ A)$ is purely inseparable. This proves that $D$ is local and that the ring maps $B \to D$ and $C \to D$ are local. $\square$

Lemma 10.156.6. Let $A \to B$ and $A \to C$ be ring maps. Let $\kappa$ be a separably algebraically closed field and let $B \otimes _ A C \to \kappa$ be a ring homomorphism. Denote

$\xymatrix{ B^{sh} \ar[r] & (B \otimes _ A C)^{sh} \\ A^{sh} \ar[u] \ar[r] & C^{sh} \ar[u] }$

the corresponding maps of strict henselizations (see proof). If

1. $A \to B$ is quasi-finite at the prime $\mathfrak p_ B = \mathop{\mathrm{Ker}}(B \to \kappa )$, or

2. $B$ is a filtered colimit of quasi-finite $A$-algebras, or

3. $B_{\mathfrak p_ B}$ is a filtered colimit of quasi-finite algebras over $A_{\mathfrak p_ A}$, or

4. $B$ is integral over $A$,

then $B^{sh} \otimes _{A^{sh}} C^{sh} \to (B \otimes _ A C)^{sh}$ is an isomorphism.

Proof. Write $D = B \otimes _ A C$. Denote $\mathfrak p_ A = \mathop{\mathrm{Ker}}(A \to \kappa )$ and similarly for $\mathfrak p_ B$, $\mathfrak p_ C$, and $\mathfrak p_ D$. Denote $\kappa _ A \subset \kappa$ the separable algebraic closure of $\kappa (\mathfrak p_ A)$ in $\kappa$ and similarly for $\kappa _ B$, $\kappa _ C$, and $\kappa _ D$. Denote $A^{sh}$ the strict henselization of $A_{\mathfrak p_ A}$ constructed using the separable algebraic closure $\kappa _ A/\kappa (\mathfrak p_ A)$. Similarly for $B^{sh}$, $C^{sh}$, and $D^{sh}$. We obtain the commutative diagram of the lemma from the functoriality of Lemma 10.155.10.

Consider the map

$c : B^{sh} \otimes _{A^{sh}} C^{sh} \to D^{sh} = (B \otimes _ A C)^{sh}$

we obtain from the commutative diagram. If $A \to B$ is quasi-finite at $\mathfrak p_ B = \mathop{\mathrm{Ker}}(B \to \kappa )$, then the ring map $C \to D$ is quasi-finite at $\mathfrak p_ D$ by Lemma 10.122.6. Hence by Lemma 10.156.3 (and Lemma 10.36.13) the ring map $c$ is a homomorphism of finite $C^{sh}$-algebras and

$B^{sh} = (B \otimes _ A A^{sh})_{\mathfrak q} \quad \text{and}\quad D^{sh} = (D \otimes _ C C^{sh})_{\mathfrak r} = (B \otimes _ A C^{sh})_{\mathfrak r}$

for some primes $\mathfrak q$ and $\mathfrak r$. Since

$B^{sh} \otimes _{A^{sh}} C^{sh} = (B \otimes _ A A^{sh})_{\mathfrak q} \otimes _{A^{sh}} C^{sh} = \text{a localization of } B \otimes _ A C^{sh}$

we conclude that source and target of $c$ are both localizations of $B \otimes _ A C^{sh}$ (compatibly with the map). Hence it suffices to show that $B^{sh} \otimes _{A^{sh}} C^{sh}$ is local (small detail omitted). This follows from Lemma 10.156.5 and the fact that $A^{sh} \to B^{sh}$ is finite with purely inseparable residue field extension by the already used Lemma 10.156.3. This proves case (1) of the lemma.

In case (2) write $B = \mathop{\mathrm{colim}}\nolimits B_ i$ as a filtered colimit of quasi-finite $A$-algebras. We correspondingly get $D = \mathop{\mathrm{colim}}\nolimits D_ i$ with $D_ i = B_ i \otimes _ A C$. Observe that $B^{sh} = \mathop{\mathrm{colim}}\nolimits B_ i^{sh}$. Namely, the ring $\mathop{\mathrm{colim}}\nolimits B_ i^{sh}$ is a strictly henselian local ring by Lemma 10.154.8. Also $\mathop{\mathrm{colim}}\nolimits B_ i^{sh}$ is a filtered colimit of étale $B$-algebras by Lemma 10.154.4. Finally, the residue field of $\mathop{\mathrm{colim}}\nolimits B_ i^{sh}$ is a separable algebraic closure of $\kappa (\mathfrak p_ B)$ (details omitted). Hence we conclude that $B^{sh} = \mathop{\mathrm{colim}}\nolimits B_ i^{sh}$, see discussion following Definition 10.155.3. Similarly, we have $D^{sh} = \mathop{\mathrm{colim}}\nolimits D_ i^{sh}$. Then we conclude by case (1) because

$D^{sh} = \mathop{\mathrm{colim}}\nolimits D_ i^{sh} = \mathop{\mathrm{colim}}\nolimits B_ i^{sh} \otimes _{A^{sh}} C^{sh} = B^{sh} \otimes _{A^{sh}} C^{sh}$

since filtered colimit commute with tensor products.

Case (3). We may replace $A$, $B$, $C$ by their localizations at $\mathfrak p_ A$, $\mathfrak p_ B$, and $\mathfrak p_ C$. Thus (3) follows from (2).

Since an integral ring map is a filtered colimit of finite ring maps, we see that (4) follows from (2) as well. $\square$

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