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

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. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/


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 µ l the étale sheaf of l -th roots of unity. Then the sequence of étale cohomology groups is exact by Bloch-Ogus ( [2]) and Panin ([10]). Here κ(p) is the residue field of p ∈ 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 ≥ 1 be an integer and l a positive integer which is invertible in R. Is the natural map injective ?
Here Ω K is the set of normalized discrete valuations on K and K v is the corresponding henselization of K for each v ∈ Ω 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 µ ⊗n l coefficients holds over 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 , µ ⊗n l ) holds for n ≥ 1. That is, the local-global map (2) is injective for n ≥ 1. V. Suresh also proved Theorem 2 by an alternative method (cf. [8, Remark in Theorem 1.2]).

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 κ(p) is the residue field of p ∈ X . If X = Spec R, k(Spec R) is abbreviated as k(R). The symbol µ l denotes the étale sheaf of l -th roots of unity.

Proof of the main result (Theorem 1)
In this section, we use the following results (Theorem 3 and Theorem 4) repeatedly:  In this section, we use Theorem 4 in the case where dim X ≤ 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 ∉ m. Then the homomorphism
is injective for any i ≥ 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 ∈ m \ m 2 and p = (a). Then R/p is a henselian regular local ring of dimension d and is commutative. Then the left vertical map in the diagram (4) 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 is exact for any i ≥ 0.
Proof. Let A be a Dedekind ring, q a maximal ideal of A. Then by Theorem 4. Hence the sequence by [9, pp. 88-89, III, Lemma 1.16], the sequence is exact.
Let m be the maximal ideal of R. Let g ∈ m\m 2 , p = (g ) and Z = Spec R/p. Then R/p is a regular local ring and we have by Theorem 4. We consider the commutative diagram where and 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 and R/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. This completes the proof.