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

TL;DR

This paper proves Gersten's conjecture for étale cohomology over two-dimensional henselian regular local rings without equi-characteristic assumptions, and applies it to establish a local-global principle for Galois cohomology in mixed characteristic cases.

Contribution

It extends the validity of Gersten's conjecture to mixed characteristic two-dimensional henselian regular local rings and derives a local-global principle for Galois cohomology.

Findings

Gersten's conjecture holds for étale cohomology in the specified setting.

Established a local-global principle for Galois cohomology in mixed characteristic.

No equi-characteristic assumption needed for the main result.

Abstract

We show the Gersten's conjecture for \'etale cohomology over two dimensional henselian regular local rings without assuming equi-characteristic. As application, we obtain the local-global principle for Galois cohomology over mixed characteristic two-dimensional henselian local rings.

Equations

$$0\to H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}\left(R,{\mu }_l^{\otimes n}\right)\to$$

$$\to$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}\left(K,{\mu }_l^{\otimes n}\right)\to \prod _{v\in {\Omega }_K}{H}_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}\left(K_v,{\mu }_l^{\otimes n}\right)$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(A,{\mu }_l^{\otimes n})\to H_{\acute{\mathrm{e}}\mathrm{t}}^i(K,{\mu }_l^{\otimes n})$$

$${\Lambda }_Y\stackrel{\sim }{\to }R{i}^{!}\Lambda (c)[2c]$$

$$H_{\acute{e}t}^i\left(\mathrm{Spec}R,{\mu }_l^{\otimes n}\right)\to H_{\acute{e}t}^i\left(\mathrm{Spec}K,{\mu }_l^{\otimes n}\right)$$

$$k(R/\mathfrak{p})=R_{\mathfrak{p}}/\mathfrak{p}{R}_{\mathfrak{p}}$$

$$\begin{array}{ccc}{H}_{\acute{e}t}^i\left(\mathrm{Spec}R,{\mu }_l^{\otimes n}\right)& \to & H_{\acute{e}t}^i\left(\mathrm{Spec}{R}_{\mathfrak{p}},{\mu }_l^{\otimes n}\right)\\ \downarrow & & \downarrow \\ H_{\acute{e}t}^i\left(\mathrm{Spec}R/\mathfrak{p},{\mu }_l^{\otimes n}\right)& \to & H_{\acute{e}t}^i\left(\mathrm{Spec}k(R/\mathfrak{p}),{\mu }_l^{\otimes n}\right)\end{array}$$

$$H_{\acute{e}t}^i\left(\mathrm{Spec}R,{\mu }_l^{\otimes n}\right)\to H_{\acute{e}t}^i\left(\mathrm{Spec}{R}_{\mathfrak{p}},{\mu }_l^{\otimes n}\right)$$

$$H_{\acute{e}t}^i\left(\mathrm{Spec}{R}_{\mathfrak{p}},{\mu }_l^{\otimes n}\right)\to H_{\acute{e}t}^i\left(\mathrm{Spec}K,{\mu }_l^{\otimes n}\right)$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(R,{\mu }_l^{\otimes n})\to H_{\acute{\mathrm{e}}\mathrm{t}}^i(k(R),{\mu }_l^{\otimes n})\stackrel{(\ast )}{\to }\underset{\begin{array}{c}\mathfrak{p}\in {\left(\mathrm{Spec}R\right)}^{(1)}\end{array}}{⨁}{H}_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(\kappa (\mathfrak{p}),{\mu }_l^{\otimes (n-1)})$$

$$H_{\mathfrak{q}}^{i+1}((\mathrm{Spec}A{)}_{\acute{\mathrm{e}}\mathrm{t}},{\mu }_l^{\otimes n})=H_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(\kappa (\mathfrak{q}),{\mu }_l^{\otimes (n-1)})$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(A,{\mu }_l^{\otimes n})\to H_{\acute{\mathrm{e}}\mathrm{t}}^i(U,{\mu }_l^{\otimes n})\to \underset{\mathfrak{q}\in Z^{(1)}}{⨁}{H}_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(\kappa (\mathfrak{q}),{\mu }_l^{\otimes (n-1)})$$

$$\underset{\begin{array}{c}\to \\ U\end{array}}{\lim }{H}_{\acute{\mathrm{e}}\mathrm{t}}^i(U,{\mu }_l^{\otimes n})=H_{\acute{\mathrm{e}}\mathrm{t}}^i(k(A),{\mu }_l^{\otimes n})$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(A,{\mu }_l^{\otimes n})\to H_{\acute{\mathrm{e}}\mathrm{t}}^i(k(A),{\mu }_l^{\otimes n})\to \underset{\begin{array}{c}\mathfrak{q}\in {\left(\mathrm{Spec}A\right)}^{(1)}\end{array}}{⨁}{H}_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(\kappa (\mathfrak{q}),{\mu }_l^{\otimes (n-1)})$$

$$H_Z^{i+1}((\mathrm{Spec}R{)}_{\acute{\mathrm{e}}\mathrm{t}},{\mu }_l^{\otimes n})=H_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(R/\mathfrak{p},{\mu }_l^{\otimes (n-1)})$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(R,{\mu }_l^{\otimes n})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(R/\mathfrak{p},{\mu }_l^{\otimes (n-1)}{)}^{\prime }=\mathrm{Im}\left(H_{\acute{\mathrm{e}}\mathrm{t}}^i(R_g,{\mu }_l^{\otimes n})\to H_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(R/\mathfrak{p},{\mu }_l^{\otimes (n-1)})\right)$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^i(R_g,{\mu }_l^{\otimes n}{)}^{\prime }$$

$$=$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(R/\mathfrak{p},{\mu }_l^{\otimes (n-1)}{)}^{\prime }\subset H_{\acute{\mathrm{e}}\mathrm{t}}^{i-1}(R/\mathfrak{p},{\mu }_l^{\otimes (n-1)})$$

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

$$H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}(R,{\mu }_l^{\otimes n})\to H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}(R_g,{\mu }_l^{\otimes n})$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^n(R_g,{\mu }_l^{\otimes n})\to H_{\acute{\mathrm{e}}\mathrm{t}}^{n-1}(R/\mathfrak{p},{\mu }_l^{\otimes n-1})$$

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

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

$$\begin{array}{ccc}{H}_{\mathrm{Zar}}^{n+1}(R_g,\mathbb{Z}/l(n))& \to & H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}(R_g,\mathbb{Z}/l(n))\\ \downarrow & & \downarrow \\ H_{\mathrm{Zar}}^{n+1}(k(R_g),\mathbb{Z}/l(n))& \to & H_{\acute{\mathrm{e}}\mathrm{t}}^{n+1}(k(R_g),\mathbb{Z}/l(n)).\end{array}$$

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

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

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

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

Full text

Indian Institute of Science Education and Research, Mohali111 Present affiliation: Indian Institute of Science Education and Research, Bhopal.

E-mail address: makoto@iiserb.ac.in, sakagaito43@gmail.com

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

[TABLE]

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

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

[TABLE]

injective ?

Here $\Omega_{K}$ is the set of normalized discrete valuations on $K$ and $K_{v}$ is the corresponding henselization of $K$ for each $v\in\Omega_{K}$.

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 1.1**.**

(Theorem 2.7)

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 $\operatorname{Spec}R$. That is, the sequence (1) is exact.

See Remark 2.6 (iii) for the reason why we assume $\operatorname{dim}(R)=2$ in Theorem 1.1. We obtain the following result as an application of Theorem 1.1 :

Theorem 1.2**.**

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 $\operatorname{H}^{n+1}(K,\mu_{l}^{\otimes n})$ holds for $n\geq 1$. That is, the local-global map (2) is injective for $n\geq 1$.

V.Suresh also proved Theorem 1.2 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 $\kappa(\mathfrak{p})$ is the residue field of $\mathfrak{p}\in X$. If $X=\operatorname{Spec}R$, $k(\operatorname{Spec}R)$ 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 1.1)

In this section, we use the following results (Theorem 2.1 and Theorem 2.2) repeatedly:

Theorem 2.1**.**

(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

[TABLE]

is injective for any $i\geq 0$.

Theorem 2.2**.**

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

[TABLE]

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

In this section, we use Theorem 2.2 in the case where $\operatorname{dim}X\leq 2$. In this case, Theorem 2.2 was proved much earlier by Gabber in 1976. See also [11, §5, Remark 5.6] for a published proof.

Proposition 2.3**.**

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

[TABLE]

is injective for any $i\geq 0$.

Proof.

We prove the statement by induction on $\operatorname{dim}(R)$. Let $R$ be a discrete valuation ring (which does not need to be henselian). Then the homomorphism (3) is injective by Theorem 2.1.

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

[TABLE]

where $k(R/\mathfrak{p})$ is the function field of $R/\mathfrak{p}$.

Therefore the diagram

[TABLE]

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

[TABLE]

is injective. Moreover the homomorphism

[TABLE]

is injective by Theorem 2.1. Therefore the statement follows. ∎

Proposition 2.4**.**

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

[TABLE]

is exact for any $i\geq 0$.

Proof.

Let $A$ be a Dedekind ring, $\mathfrak{q}$ a maximal ideal of $A$. Then

[TABLE]

by Theorem 2.2. Hence the sequence

[TABLE]

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

[TABLE]

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

[TABLE]

is exact.

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

[TABLE]

by Theorem 2.2.

We consider the commutative diagram

[TABLE]

where

[TABLE]

and

[TABLE]

Then the rows in the diagram (6) are exact by Theorem 2.2. Since $R_{g}$ is a Dedekind domain, the middle map in the diagram (6) is surjective by (5). Moreover, since

[TABLE]

and $R/\mathfrak{p}$ is a discrete valuation ring, the right map in the diagram (6) is injective by Theorem 2.1. Therefore the statement follows from the snake lemma. ∎

Corollary 2.5**.**

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 $\operatorname{dim}(R)=2$. Then

[TABLE]

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

Proof.

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

[TABLE]

is injective by Proposition 2.3. Hence the homomorphism

[TABLE]

is surjective by Theorem 2.2. Therefore the homomorphism

[TABLE]

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

[TABLE]

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

[TABLE]

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 2.4. Hence the homomorphism

[TABLE]

is injective and the homomorphism

[TABLE]

is also injective. Since

[TABLE]

we have

[TABLE]

This completes the proof. ∎

Remark 2.6**.**

• (i)

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

[TABLE]

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

• (ii)

If we have

[TABLE]

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

[TABLE]

is injective by a similar augument as in the proof of **[13, Theorem 4.2]**.

• (iii)

The reason why we assume $\operatorname{dim}(R)=2$ in Propositin 2.4 and Theorem 2.7 is that we have to show that the middle map in the diagram (6), i.e., the homomorphism

[TABLE]

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

[TABLE]

If we have

[TABLE]

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

[TABLE]

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

[TABLE]

and Proposition 2.4 holds. Here $\epsilon:(\operatorname{Spec}R_{g})_{\mathrm{\acute{e}t}}\to(\operatorname{Spec}R_{g})_{\mathrm{Zar}}$ is the change of site maps.

Theorem 2.7**.**

Let $R$ be a henselian regular local ring with $\operatorname{dim}(R)=2$ and $l$ a positive integer which is invertible in $R$. Then the sequence

[TABLE]

is exact for any $i\geq 0$.

Proof.

The exactness of the complex (2.7) at the first two terms follows from Proposition 2.3 and Proposition 2.4.

We consider the coniveau spectral sequence

[TABLE]

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

[TABLE]

such that

[TABLE]

By Theorem 2.2, it suffices to show that

[TABLE]

By Proposition 2.3, the morphism

[TABLE]

is injective and

[TABLE]

Hence we have

[TABLE]

Since

[TABLE]

for $p\geq 3$ and

[TABLE]

for $r\geq 2$, we have

[TABLE]

By the exactness of the complex (2.7) at the second term, we have

[TABLE]

and

[TABLE]

Hence we have

[TABLE]

Moreover, since

[TABLE]

for $r\geq 3$, we have

[TABLE]

for $r\geq 3$. Therefore

[TABLE]

This completes the proof. ∎