# Comptes Rendus
## Mathématique

Makoto Sakagaito

A note on Gersten's conjecture for étale cohomology over
two-dimensional henselian regular local rings

Volume 358, issue 1 (2020), p. 33-39

Published online: 18 March 2020

https://doi.org/10.5802/crmath.9

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CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.

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Number Theory, Homological Algebra / *Théorie des nombres, Algèbre homologique*

# A note on Gersten's conjecture for étale cohomology over two-dimensional henselian regular local rings

*Une note sur la conjecture de Gersten pour la cohomologie étale sur des anneaux locaux réguliers henséliens à deux dimensions*

Makoto Sakagaito<sup>a</sup>

**Abstract.** We prove Gersten's conjecture for étale cohomology over two dimensional henselian regular local rings without assuming equi-characteristic. As an application, we obtain the local-global principle for Galois cohomology over mixed characteristic two-dimensional henselian local rings.

**Résumé.** Nous montrons la conjecture de Gersten pour la cohomologie étale sur des anneaux locaux réguliers henséliens sans supposer de caractère équicaractéristique. En application, nous obtenons le principe local-global pour la cohomologie de Galois sur des anneaux locaux henséliens à deux dimensions de caractéristique mixte.

Manuscript received 30th April 2019, revised 17th September 2019 and 24th November 2019, accepted 17th December 2019.

<footnote><sup>a</sup> Indian Institute of Science Education and Research, Mohali.</footnote>

<footnote>*E-mail:* makoto@iiserb.ac.in.</footnote>
# 1. Introduction

Let $R$ be an equi-characteristic regular local ring, $k(R)$ the field of fractions of $R$, $l$ a positive integer which is invertible in $R$ and $\mu_l$ the étale sheaf of $l$-th roots of unity. Then the sequence of étale cohomology groups

$$ 
\begin{aligned}
0 \to H_{\text{ét}}^{n+1}(R, \mu_l^{\otimes n}) &\to H_{\text{ét}}^{n+1}(k(R), \mu_l^{\otimes n}) \quad (1) \\
&\to \bigoplus_{\substack{p \in \text{Spec } R \\ \text{ht}(p)=1}} H_{\text{ét}}^n(\kappa(p), \mu_l^{\otimes (n-1)}) \\
&\to \bigoplus_{\substack{p \in \text{Spec } R \\ \text{ht}(p)=2}} H_{\text{ét}}^{n-1}(\kappa(p), \mu_l^{\otimes (n-2)}) \to \dots
\end{aligned}
$$

is exact by Bloch-Ogus ([2]) and Panin ([10]). Here $\kappa(p)$ is the residue field of $p \in \text{Spec } R$.

By using the exactness of the complex (1) at the first two terms, Harbater-Hartmann-Krashen ([7]) and Hu ([8]) proved the local-global principle as follows.

Let $K$ be a field of one of the following types:

(a) (semi-global case) The function field of a connected regular projective curve over the field of fractions of a henselian excellent discrete valuation ring $A$.

(b) (local case) The function field of a two-dimensional henselian excellent normal local domain $A$.

Then the following question was raised by Colliot-Thélène ([3]):

Let $n \ge 1$ be an integer and $l$ a positive integer which is invertible in $R$. Is the natural map

$$ H_{\text{ét}}^{n+1}(K, \mu_l^{\otimes n}) \to \prod_{\nu \in \Omega_K} H_{\text{ét}}^{n+1}(K_\nu, \mu_l^{\otimes n}) \quad (2) $$

injective?

Here $\Omega_K$ is the set of normalized discrete valuations on $K$ and $K_\nu$ is the corresponding henselization of $K$ for each $\nu \in \Omega_K$.

Suppose that $A$ is equi-characteristic. Harbater-Hartmann-Krashen ([7, Theorem 3.3.6]) proved that the local-global map (2) is injective in the semi-global case. Later, Hu ([8, Theorem 2.5]) proved that the local-global map (2) is injective in both the semi-global case and the local case by an alternative method.

If the sequence (1) is exact (at the first two terms) in the case where $R$ is a mixed characteristic two-dimensional excellent henselian local ring, then the local-global map (2) is injective even without assuming equi-characteristic (cf. [7, Remark 3.3.7] and [8, Remark 2.6 (2)]).

In the case where $R$ is a local ring of a smooth algebra over a (mixed characteristic) discrete valuation ring, the sequence (1) is exact (cf. [6, Theorem 1.2 and Theorem 3.2 b)]).

In this paper, we show the following result:

**Theorem 1 (Theorem 9).** Let $R$ be a mixed characteristic two-dimensional excellent henselian local ring and $l$ a positive integer which is invertible in $R$. Then Gersten's conjecture for étale cohomology with $\mu_l^{\otimes n}$ coefficients holds over $\text{Spec } R$. That is, the sequence (1) is exact.

See Remark 8(iii) for the reason why we assume $\dim(R) = 2$ in Theorem 1. We obtain the following result as an application of Theorem 1:

**Theorem 2.** With notations as above, assume that $A$ is mixed characteristic and $l$ is a positive integer which is invertible in $A$.

In both the semi-global case and the local case, the local-global principle for the Galois cohomology group $H^{n+1}(K, \mu_l^{\otimes n})$ holds for $n \ge 1$. That is, the local-global map (2) is injective for $n \ge 1$.

V. Suresh also proved Theorem 2 by an alternative method (cf. [8, Remark in Theorem 1.2]).
### 1.1. Notations

For a scheme $X$, $X^{(i)}$ is the set of points of codimension $i$, $k(X)$ is the ring of rational functions on $X$ and $\kappa(p)$ is the residue field of $p \in X$. If $X = \text{Spec } R$, $k(\text{Spec } R)$ is abbreviated as $k(R)$. The symbol $\mu_l$ denotes the étale sheaf of $l$-th roots of unity.

## 2. Proof of the main result (Theorem 1)

In this section, we use the following results (Theorem 3 and Theorem 4) repeatedly:

**Theorem 3 (cf. [4, Theorem B.2.1 and Examples B.1.1.(2)])**. Let $A$ be a discrete valuation ring, $K$ the function field of $A$ and $l$ a positive integer which is invertible in $A$. Then the homomorphism

$$H_{\text{ét}}^{i}(A, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i}(K, \mu_{l}^{\otimes n})$$

is injective for any $i \ge 0$.

**Theorem 4 (The absolute purity theorem [5, p. 159, Theorem 2.1.1])**. Let $Y \xrightarrow{i} X$ be a closed immersion of noetherian regular schemes of pure codimension $c$. Let $n$ be an integer which is invertible on $X$, and let $\Lambda = \mathbb{Z}/n$. Then the cycle class (cf. [5, 1.1]) give an isomorphism

$$\Lambda_Y \xrightarrow{\sim} R i^! \Lambda(c) [2c]$$

in $D^{+}(Y_{\text{ét}}, \Lambda)$. Here $D^{+}(Y_{\text{ét}}, \Lambda)$ is the derived category of complexes bounded below of étale sheaves of $\Lambda$-modules on $Y$.

In this section, we use Theorem 4 in the case where $\dim X \le 2$. In this case, Theorem 4 was proved much earlier by Gabber in 1976. See also [11, §5, Remark 5.6] for a published proof.

**Proposition 5.** Let $R$ be a henselian regular local ring, $m$ the maximal ideal of $R$ and $K$ the function field of $R$. Let $l$ be a positive integer such that $l \notin \mathbb{m}$. Then the homomorphism

$$H_{\text{ét}}^{i}(\text{Spec } R, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i}(\text{Spec } K, \mu_{l}^{\otimes n}) \quad (3)$$

is injective for any $i \ge 0$.

**Proof.** We prove the statement by induction on $\dim(R)$. Let $R$ be a discrete valuation ring (which does not need to be henselian). Then the homomorphism (3) is injective by Theorem 3.

Assume that the statement is true for a henselian regular local ring of dimension $d$.

Let $R$ be a henselian regular local ring of dimension $d+1$, $a \in \mathbb{m} \setminus \mathbb{m}^2$ and $p = (a)$. Then $R/p$ is a henselian regular local ring of dimension $d$ and

$$k(R/p) = R_p/pR_p$$

where $k(R/p)$ is the function field of $R/p$.

Therefore the diagram

$$\begin{tikzcd}
& & H_{\text{ét}}^{i}(\text{Spec } R, \mu_{l}^{\otimes n}) \arrow[r] \arrow[d] & H_{\text{ét}}^{i}(\text{Spec } R_{p}, \mu_{l}^{\otimes n}) \arrow[d] \\
\text{Spec } R/p & \text{Spec } K/p & \text{Spec } k(R/p) & \text{Spec } \mathbb{Z}/p
\end{tikzcd} \quad (4)$$

is commutative. Then the left vertical map in the diagram (4) is an isomorphism by [1, p. 93, Theorem (4.9)] and the bottom horizontal map in the diagram (4) is injective by the induction hypothesis. Hence the homomorphism

$$H_{\text{ét}}^{i}(\text{Spec } R, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i}(\text{Spec } R_{p}, \mu_{l}^{\otimes n})$$
is injective. Moreover the homomorphism

$$H_{\text{ét}}^{i}(\text{Spec } R_{\mathfrak{p}}, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i}(\text{Spec } K, \mu_{l}^{\otimes n})$$

is injective by Theorem 3. Therefore the statement follows. □

**Proposition 6 (cf. [12, Proposition 4.7]).** Let $R$ be a regular local ring and $l$ a positive integer which is invertible in $R$. Suppose that $\dim(R) = 2$. Then the sequence

$$H_{\text{ét}}^{i}(R, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i}(k(R), \mu_{l}^{\otimes n}) \xrightarrow{(*)} \bigoplus_{\mathfrak{p} \in (\text{Spec } R)^{(1)}} H_{\text{ét}}^{i-1}(\kappa(\mathfrak{p}), \mu_{l}^{\otimes(n-1)})$$

is exact for any $i \ge 0$.

**Proof.** Let $A$ be a Dedekind ring, $q$ a maximal ideal of $A$. Then

$$H_{q}^{i+1}((\text{Spec } A)_{\text{ét}}, \mu_{l}^{\otimes n}) = H_{\text{ét}}^{i-1}(\kappa(q), \mu_{l}^{\otimes(n-1)})$$

by Theorem 4. Hence the sequence

$$H_{\text{ét}}^{i}(A, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i}(U, \mu_{l}^{\otimes n}) \to \bigoplus_{q \in Z^{(1)}} H_{\text{ét}}^{i-1}(\kappa(q), \mu_{l}^{\otimes(n-1)})$$

is exact where $Z$ is a closed subscheme of $\text{Spec } A$ and $U = \text{Spec } R \setminus Z$. Since

$$\lim_{\underset{U}{\longrightarrow}} H_{\text{ét}}^{i}(U, \mu_{l}^{\otimes n}) = H_{\text{ét}}^{i}(k(A), \mu_{l}^{\otimes n})$$

by [9, pp. 88–89, III, Lemma 1.16], the sequence

$$H_{\text{ét}}^{i}(A, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i}(k(A), \mu_{l}^{\otimes n}) \to \bigoplus_{q \in (\text{Spec } A)^{(1)}} H_{\text{ét}}^{i-1}(\kappa(q), \mu_{l}^{\otimes(n-1)}) \quad (5)$$

is exact.

Let $m$ be the maximal ideal of $R$. Let $g \in m \setminus m^2$, $\mathfrak{p} = (g)$ and $Z = \text{Spec } R/\mathfrak{p}$. Then $R/\mathfrak{p}$ is a regular local ring and we have

$$H_{Z}^{i+1}((\text{Spec } R)_{\text{ét}}, \mu_{l}^{\otimes n}) = H_{\text{ét}}^{i-1}(R/\mathfrak{p}, \mu_{l}^{\otimes(n-1)})$$

by Theorem 4.

We consider the commutative diagram

$$\begin{tikzcd}
& & H_{\text{ét}}^{i}(R, \mu_{l}^{\otimes n}) \arrow[r] \arrow[d] & H_{\text{ét}}^{i}(R_{g}, \mu_{l}^{\otimes n}) \arrow[r] \arrow[d] & H_{\text{ét}}^{i-1}(R/\mathfrak{p}, \mu_{l}^{\otimes(n-1)})' \arrow[d] \arrow[r] & 0 \\
0 \arrow[r] & \text{Ker}(\ast) \arrow[r] & H_{\text{ét}}^{i}(R_{g}, \mu_{l}^{\otimes n})' \arrow[r] & H_{\text{ét}}^{i-1}(k(R/\mathfrak{p}), \mu_{l}^{\otimes(n-1)}) \arrow[r] & 0
\end{tikzcd}
\quad (6)$$

where

$$H_{\text{ét}}^{i-1}(R/\mathfrak{p}, \mu_{l}^{\otimes(n-1)})' = \text{Im} \left( H_{\text{ét}}^{i}(R_g, \mu_{l}^{\otimes n}) \to H_{\text{ét}}^{i-1}(R/\mathfrak{p}, \mu_{l}^{\otimes(n-1)}) \right)$$

and

$$H_{\text{ét}}^{i}(R_g, \mu_{l}^{\otimes n})' = \text{Ker} \left( H_{\text{ét}}^{i}(k(R_g), \mu_{l}^{\otimes n}) \to \bigoplus_{q \in (\text{Spec } R_g)^{(1)}} H_{\text{ét}}^{i-1}(\kappa(q)), \mu_{l}^{\otimes(n-1)}) \right).$$

Then the rows in the diagram (6) are exact by Theorem 4. Since $R_g$ is a Dedekind domain, the middle map in the diagram (6) is surjective by (5). Moreover, since

$$H_{\text{ét}}^{i-1}(R/\mathfrak{p}, \mu_{l}^{\otimes(n-1)})' \subset H_{\text{ét}}^{i-1}(R/\mathfrak{p}, \mu_{l}^{\otimes(n-1)})$$

and $R/\mathfrak{p}$ is a discrete valuation ring, the right map in the diagram (6) is injective by Theorem 3. Therefore the statement follows from the snake lemma. □
**Corollary 7.** Let $R$ be the henselization of a regular local ring which is essentially of finite type over a mixed characteristic discrete valuation ring. Suppose that $\dim(R) = 2$. Then

$$H_{\mathrm{Zar}}^{n+1}(R, \mathbb{Z}/l(n)) = 0$$

for a positive integer $l$ which is invertible in $R$. Here $\mathbb{Z}(n)$ is Bloch's cycle complex and $\mathbb{Z}/l(n) = \mathbb{Z}(n) \otimes \mathbb{Z}/l$ (cf. [6, p. 779]).

**Proof.** Let $m$ be the maximal ideal of $R$. Let $g \in m \setminus m^2$ and $\mathfrak{p} = (g)$. Then the homomorphism

$$H_{\mathrm{ét}}^{n+1}(R, \mu_l^{\otimes n}) \to H_{\mathrm{ét}}^{n+1}(R_g, \mu_l^{\otimes n})$$

is injective by Proposition 5. Hence the homomorphism

$$H_{\mathrm{ét}}^{n}(R_g, \mu_l^{\otimes n}) \to H_{\mathrm{ét}}^{n-1}(R/\mathfrak{p}, \mu_l^{\otimes n-1})$$

is surjective by Theorem 4. Therefore the homomorphism

$$H_{\mathrm{Zar}}^{n}(R_g, \mathbb{Z}/l(n)) \to H_{\mathrm{Zar}}^{n-1}(R/\mathfrak{p}, \mathbb{Z}/l(n-1))$$

is surjective by [6, p. 774, Theorem 1.2] and [14]. Moreover the homomorphism

$$H_{\mathrm{Zar}}^{n+1}(R, \mathbb{Z}/l(n)) \to H_{\mathrm{Zar}}^{n+1}(R_g, \mathbb{Z}/l(n))$$

is injective by the localization theorem [6, p. 779, Theorem 3.2]. We consider the commutative diagram

$$\begin{tikzcd}
H_{\mathrm{Zar}}^{n+1}(R_g, \mathbb{Z}/l(n)) \arrow[r] \arrow[d] & H_{\mathrm{ét}}^{n+1}(R_g, \mathbb{Z}/l(n)) \arrow[d] \\
H_{\mathrm{Zar}}^{n+1}(k(R_g), \mathbb{Z}/l(n)) \arrow[r] & H_{\mathrm{ét}}^{n+1}(k(R_g), \mathbb{Z}/l(n)).
\end{tikzcd} \qquad (7)$$

Then the upper map in the commutative diagram (7) is injective by the Beilinson–Lichenbaum conjecture ([6, p. 774, Theorem 1.2], [14]) and the right map in the commutative diagram (7) is injective by the commutative diagram (6) in the proof of Proposition 6. Hence the homomorphism

$$H_{\mathrm{Zar}}^{n+1}(R_g, \mathbb{Z}/l(n)) \to H_{\mathrm{Zar}}^{n+1}(k(R_g), \mathbb{Z}/l(n))$$

is injective and the homomorphism

$$H_{\mathrm{Zar}}^{n+1}(R, \mathbb{Z}/l(n)) \to H_{\mathrm{Zar}}^{n+1}(k(R_g), \mathbb{Z}/l(n))$$

is also injective. Since

$$H_{\mathrm{Zar}}^{n+1}(k(R_g), \mathbb{Z}/l(n)) = 0,$$

we have

$$H_{\mathrm{Zar}}^{n+1}(R, \mathbb{Z}/l(n)) = 0.$$

This completes the proof. $\square$

**Remark 8.**

(i) If $R$ is a local ring of a smooth algebra over a discrete valuation ring, then

$$H_{\mathrm{Zar}}^{i}(R, \mathbb{Z}/m(n)) = 0$$

for $i > n$ and any positive integer $m$ (cf. [6, p. 786, Corollary 4.4]).

(ii) If we have

$$H_{\mathrm{Zar}}^{n+1}(R, \mathbb{Z}/l(n)) = 0$$

for any regular local ring $R$ which is finite type over a discrete valuation ring and a positive integer $l$ which is invertible in $R$, we can show that the homomorphism

$$H_{\mathrm{ét}}^{n+1}(R, \mu_l^{\otimes n}) \to H_{\mathrm{ét}}^{n+1}(k(R), \mu_l^{\otimes n})$$

is injective by a similar argument as in the proof of [13, Theorem 4.2].
(iii) The reason why we assume $dim(R) = 2$ in Proposition 6 and Theorem 9 is that we have to show that the middle map in the diagram (6), i.e., the homomorphism

$$H_{\text{ét}}^{n+1}(R_g, \mu_l^{\otimes n}) \to H_{\text{ét}}^{n+1}(R_g, \mu_l^{\otimes n})'$$

is surjective for an element $g$ of $\mathfrak{m} \setminus \mathfrak{m}^2$. Here $\mathfrak{m}$ is the maximal ideal of $R$ and

$$H_{\text{ét}}^{n+1}(R_g, \mu_l^{\otimes n})' = \text{Ker} \left( H_{\text{ét}}^{n+1}(k(R), \mu_l^{\otimes n}) \to \bigoplus_{q \in (\text{Spec } R_g)^{(1)}} H_{\text{ét}}^n(\kappa(q), \mu_l^{\otimes (n-1)}) \right).$$

If we have

$$H_{\text{Zar}}^{n+1}(R, \mathbb{Z}/l(n)) = H_{\text{Zar}}^{n+2}(R, \mathbb{Z}/l(n)) = 0$$

for any regular local ring $R$ which is finite type over a discrete valuation ring and a positive integer $l$ which is invertible in $R$, then

$$H_{\text{Zar}}^{n+1}(R_g, \mathbb{Z}/l(n)) = H_{\text{Zar}}^{n+2}(R_g, \mathbb{Z}/l(n)) = 0$$

by the localization theorem ([6, p. 779, Theorem 3.2]) and we can show that

$$H_{\text{ét}}^{n+1}(R_g, \mu_l^{\otimes n}) = \Gamma(\text{Spec } R_g, R^{n+1}\epsilon_*(\mu_l^{\otimes n})) = H_{\text{ét}}^{n+1}(R_g, \mu_l^{\otimes n})'$$

and Proposition 6 holds. Here $\epsilon: (\text{Spec } R_g)_{\text{ét}} \to (\text{Spec } R_g)_{\text{Zar}}$ is the change of site maps.

**Theorem 9.** Let $R$ be a henselian regular local ring with $dim(R) = 2$ and $l$ a positive integer which is invertible in $R$. Then the sequence

$$0 \to H_{\text{ét}}^i(R, \mu_l^{\otimes n}) \to H_{\text{ét}}^{i}(k(R), \mu_l^{\otimes n}) \to \bigoplus_{p \in (\text{Spec } R)^{(1)}} H_{\text{ét}}^{i-1}(\kappa(p), \mu_l^{\otimes (n-1)}) \to \bigoplus_{p \in (\text{Spec } R)^{(2)}} H_{\text{ét}}^{i-2}(\kappa(p), \mu_l^{\otimes (n-2)}) \to 0 \quad (8)$$

is exact for any $i \ge 0$.

**Proof.** The exactness of the complex (8) at the first two terms follows from Proposition 5 and Proposition 6.

We consider the coniveau spectral sequence

$$H_1^{p,q} = \coprod_{x \in (\text{Spec } R)^{(p)}} H_x^{p+q}(\text{Spec } R, \mu_l^{\otimes n}) \Rightarrow H_{\text{ét}}^{p+q}(R, \mu_l^{\otimes n}) = H^{p+q}$$

(cf. [4, §1]). Then we have a filtration

$$0 \subset H_{p+q}^{p+q} \subset \dots \subset H_1^{p+q} \subset H_0^{p+q} = H^{p+q},$$

such that

$$H_p^{p+q} / H_{p+1}^{p+q} \cong H_\infty^{p,q}.$$

By Theorem 4, it suffices to show that

$$H_2^{1,i-1} = H_2^{2,i-2} = 0.$$

By Proposition 5, the morphism

$$H^i \to H_\infty^{0,i}$$

is injective and

$$H_1^i = H_2^i = 0.$$

Hence we have

$$H_\infty^{1,i-1} = H_\infty^{2,i-2} = 0.$$

Since

$$H_r^{p,i-p+1} = 0$$
for $p \ge 3$ and

$$H_r^{1-r, i+r-2} = 0$$

for $r \ge 2$, we have

$$H_2^{1,i-1} = H_\infty^{1,i-1} = 0.$$

By the exactness of the complex (8) at the second term, we have

$$H_2^{0,i-1} = H_\infty^{0,i-1} = H^{i-1}$$

and

$$\mathrm{Im} \left( H_2^{0,i-1} \xrightarrow{d_2^{0,i-1}} H_2^{2,i-2} \right) = 0.$$

Hence we have

$$H_2^{2,i-2} = H_3^{2,i-2}.$$

Moreover, since

$$H_r^{2-r, i+r-3} = 0$$

for $r \ge 3$, we have

$$H_{r+1}^{2,i-2} = \frac{\mathrm{Ker}(d_r^{2,i-2})}{\mathrm{Im}(d_r^{2-r,i+r-3})} = H_r^{2,i-2}$$

for $r \ge 3$. Therefore

$$H_2^{2,i-2} = H_3^{2,i-2} = H_\infty^{2,i-2} = 0.$$

This completes the proof. □

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