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.Comment: 9 pages, Final version, Revised after the referee repor

Sakagaito, Makoto

Comptes Rendus Mathématique

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arXiv

Makoto Sakagaito

Comptes RendusMathématiqueMakoto SakagaitoA note on Gersten’s conjecture for étale cohomology overtwo-dimensional henselian regular local ringsVolume 358, issue 1 (2020), p. 33-39.<https://doi.org/10.5802/crmath.9>© Académie des sciences, Paris and the authors, 2020.Some rights reserved.This article is licensed under theCREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orgComptes RendusMathématique2020, 358, n 1, p. 33-39https://doi.org/10.5802/crmath.9Number Theory, Homological Algebra / Théorie des nombres, Algèbre homologiqueA note on Gersten’s conjecture for étalecohomology over two-dimensional henselianregular local ringsUne note sur la conjecture de Gersten pour lacohomologie étale sur des anneaux locaux réguliershenséliens à deux dimensionsMakoto Sakagaitoaa Indian Institute of Science Education and Research, Mohali.E-mail: makoto@iiserb.ac.in.Abstract. We prove Gersten’s conjecture for étale cohomology over two dimensional henselian regular localrings without assuming equi-characteristic. As an application, we obtain the local-global principle for Galoiscohomology over mixed characteristic two-dimensional henselian local rings.Résumé. Nous montrons la conjecture de Gersten pour la cohomologie étale sur des anneaux locaux régu-liers henséliens sans supposer de caractère équicaractéristique. En application, nous obtenons le principelocal-global pour la cohomologie de Galois sur des anneaux locaux henséliens à deux dimensions de carac-téristique mixte.Manuscript received 30th April 2019, revised 17th September 2019 and 24th November 2019, accepted 17th De-cember 2019.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/34 Makoto Sakagaito1. IntroductionLet R be an equi-characteristic regular local ring, k(R) the field of fractions of R, l a positiveinteger which is invertible in R and µl the étale sheaf of l -th roots of unity. Then the sequenceof étale cohomology groups0 −→ Hn+1ét(R,µ⊗nl)−→Hn+1ét (k(R),µ⊗nl ) (1)−→ ⊕p∈SpecRht(p)=1Hnét(κ(p),µ⊗(n−1)l)−→ ⊕p∈SpecRht(p)=2Hn−1ét(κ(p),µ⊗(n−2)l)−→ ·· ·is exact by Bloch–Ogus ([2]) and Panin ([10]). Here κ(p) is the residue field of p ∈ SpecR.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 thefield of fractions of a henselian excellent discrete valuation ring A.(b) (local case) The function field of a two-dimensional henselian excellent normal localdomain 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 mapHn+1ét(K ,µ⊗nl)−→ ∏v∈ΩKHn+1ét(Kv ,µ⊗nl)(2)injective ?Here ΩK is the set of normalized discrete valuations on K and Kv is the correspondinghenselization of K for each v ∈ΩK .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, Theo-rem 2.5]) proved that the local-global map (2) is injective in both the semi-global case and thelocal case by an alternative method.If the sequence (1) is exact (at the first two terms) in the case where R is a mixed characteristictwo-dimensional excellent henselian local ring, then the local-global map (2) is injective evenwithout 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) discretevaluation ring, the sequence (1) is exact (cf. [6, Theorem 1.2 and Theorem 3.2b)]).In this paper, we show the following result:Theorem 1 (Theorem 9). Let R be a mixed characteristic two-dimensional excellent henselianlocal ring and l a positive integer which is invertible in R. Then Gersten’s conjecture for étalecohomology with µ⊗nl coefficients holds over SpecR. 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 thefollowing result as an application of Theorem 1 :Theorem 2. With notations as above, assume that A is mixed characteristic and l is a positiveinteger which is invertible in A.In both the semi-global case and the local case, the local-global principle for the Galois coho-mology group Hn+1(K ,µ⊗nl ) 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]).C. R. Mathématique, 2020, 358, n 1, 33-39Makoto Sakagaito 351.1. NotationsFor a scheme X , X (i ) is the set of points of codimension i , k(X ) is the ring of rational functions onX and κ(p) is the residue field of p ∈ X . If X = SpecR, k(SpecR) is abbreviated as k(R). The symbolµl 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, Kthe function field of A and l a positive integer which is invertible in A. Then the homomorphismHiét(A,µ⊗nl ) −→ Hiét(K ,µ⊗nl )is injective for any i ≥ 0.Theorem 4 (The absolute purity theorem [5, p. 159, Theorem 2.1.1]). Let Yi,→ X be a closedimmersion of noetherian regular schemes of pure codimension c. Let n be an integer which isinvertible on X , and let Λ= Z/n. Then the cycle class (cf. [5, 1.1]) give an isomorphismΛY∼−→ Ri !Λ(c)[2c]in D+(Yét,Λ). Here D+(Yét,Λ) is the derived category of complexes bounded below of étale sheavesof Λ-modules on Y .In this section, we use Theorem 4 in the case where dim X ≤ 2. In this case, Theorem 4 wasproved 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 functionfield of R. Let l be a positive integer such that l ∉m. Then the homomorphismHiét(SpecR,µ⊗nl)−→ Hiét (SpecK ,µ⊗nl ) (3)is injective for any i ≥ 0.Proof. We prove the statement by induction on dim(R). Let R be a discrete valuation ring (whichdoes 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\m2 and p= (a). Then R/p is ahenselian regular local ring of dimension d andk(R/p) = Rp/pRpwhere k(R/p) is the function field of R/p.Therefore the diagramHiét(SpecR,µ⊗nl) //Hiét(SpecRp,µ⊗nl)Hiét(SpecR/p,µ⊗nl) // Hiét (Speck(R/p),µ⊗nl )(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 inductionhypothesis. Hence the homomorphismHiét(SpecR,µ⊗nl)−→ Hiét (SpecRp,µ⊗nl )C. R. Mathématique, 2020, 358, n 1, 33-3936 Makoto Sakagaitois injective. Moreover the homomorphismHiét(SpecRp,µ⊗nl)−→ Hiét (SpecK ,µ⊗nl )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 whichis invertible in R. Suppose that dim(R) = 2. Then the sequenceHiét(R,µ⊗nl ) −→ Hiét(k(R),µ⊗nl )(∗)−−→ ⊕p∈(SpecR)(1)Hi−1ét (κ(p),µ⊗(n−1)l )is exact for any i ≥ 0.Proof. Let A be a Dedekind ring, q a maximal ideal of A. ThenHi+1q ((Spec A)ét,µ⊗nl ) = Hi−1ét (κ(q),µ⊗(n−1)l )by Theorem 4. Hence the sequenceHiét(A,µ⊗nl ) −→ Hiét(U ,µ⊗nl ) −→⊕q∈Z (1)Hi−1ét (κ(q),µ⊗(n−1)l )is exact where Z is a closed subscheme of Spec A and U = SpecR \ Z . Sincelim→UHiét(U ,µ⊗nl ) = Hiét(k(A),µ⊗nl )by [9, pp. 88–89, III, Lemma 1.16], the sequenceHiét(A,µ⊗nl ) −→ Hiét(k(A),µ⊗nl ) −→⊕q∈(Spec A)(1)Hi−1ét (κ(q),µ⊗(n−1)l ) (5)is exact.Let m be the maximal ideal of R. Let g ∈m\m2, p= (g ) and Z = SpecR/p. Then R/p is a regularlocal ring and we haveHi+1Z ((SpecR)ét,µ⊗nl ) = Hi−1ét (R/p,µ⊗(n−1)l )by Theorem 4.We consider the commutative diagramHiét(R,µ⊗nl )//Hiét(Rg ,µ⊗nl )//Hi−1ét (R/p,µ⊗(n−1)l )′// 00 // Ker(∗) // Hiét(Rg ,µ⊗nl )′ // Hi−1ét (k(R/p),µ⊗(n−1)l )(6)whereHi−1ét (R/p,µ⊗(n−1)l )′ = Im(Hiét(Rg ,µ⊗nl ) −→ Hi−1ét (R/p,µ⊗(n−1)l ))andHiét(Rg ,µ⊗nl )′ = KerHiét(k(Rg ),µ⊗nl ) −→ ⊕q∈(SpecRg )(1)Hi−1ét (κ(q)),µ⊗(n−1)l ) .Then the rows in the diagram (6) are exact by Theorem 4. Since Rg is a Dedekind domain, themiddle map in the diagram (6) is surjective by (5). Moreover, sinceHi−1ét (R/p,µ⊗(n−1)l )′ ⊂ Hi−1ét (R/p,µ⊗(n−1)l )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. C. R. Mathématique, 2020, 358, n 1, 33-39Makoto Sakagaito 37Corollary 7. Let R be the henselization of a regular local ring which is essentially of finite type overa mixed characteristic discrete valuation ring. Suppose that dim(R) = 2. ThenHn+1Zar (R,Z/l (n)) = 0for a positive integer l which is invertible in R. Here Z(n) is Bloch’s cycle complex and Z/l (n) =Z(n)⊗Z/l (cf. [6, p. 779]).Proof. Let m be the maximal ideal of R. Let g ∈m\m2 and p= (g ). Then the homomorphismHn+1ét (R,µ⊗nl ) −→ Hn+1ét (Rg ,µ⊗nl )is injective by Proposition 5. Hence the homomorphismHnét(Rg ,µ⊗nl ) −→ Hn−1ét (R/p,µ⊗n−1l )is surjective by Theorem 4. Therefore the homomorphismHnZar(Rg ,Z/l (n)) −→ Hn−1Zar (R/p,Z/l (n −1))is surjective by [6, p. 774, Theorem 1.2] and [14]. Moreover the homomorphismHn+1Zar (R,Z/l (n)) −→ Hn+1Zar (Rg ,Z/l (n))is injective by the localization theorem [6, p. 779, Theorem 3.2]. We consider the commutativediagramHn+1Zar (Rg ,Z/l (n)) //Hn+1ét (Rg ,Z/l (n))Hn+1Zar (k(Rg ),Z/l (n)) // Hn+1ét (k(Rg ),Z/l (n)).(7)Then the upper map in the commutative diagram (7) is injective by the Beilinson–Lichenbaumconjecture ([6, p. 774, Theorem 1.2], [14]) and the right map in the commutative diagram (7) is in-jective by the commutative diagram (6) in the proof of Proposition 6. Hence the homomorphismHn+1Zar (Rg ,Z/l (n)) −→ Hn+1Zar (k(Rg ),Z/l (n))is injective and the homomorphismHn+1Zar (R,Z/l (n)) −→ Hn+1Zar (k(Rg ),Z/l (n))is also injective. SinceHn+1Zar (k(Rg ),Z/l (n)) = 0,we haveHn+1Zar (R,Z/l (n)) = 0.This completes the proof. Remark 8.(i) If R is a local ring of a smooth algebra over a discrete valuation ring, thenHiZar(R,Z/m(n)) = 0for i > n and any positive integer m (cf. [6, p. 786, Corollary 4.4]).(ii) If we haveHn+1Zar (R,Z/l (n)) = 0for any regular local ring R which is finite type over a discrete valuation ring and a positiveinteger l which is invertible in R, we can show that the homomorphismHn+1ét (R,µ⊗nl ) −→ Hn+1ét (k(R),µ⊗nl )is injective by a similar argument as in the proof of [13, Theorem 4.2].C. R. Mathématique, 2020, 358, n 1, 33-3938 Makoto Sakagaito(iii) The reason why we assume dim(R) = 2 in Propositin 6 and Theorem 9 is that we have toshow that the middle map in the diagram (6), i.e., the homomorphismHn+1ét (Rg ,µ⊗nl ) −→ Hn+1ét (Rg ,µ⊗nl )′is surjective for an element g of m\m2. Here m is the maximal ideal of R andHn+1ét (Rg ,µ⊗nl )′ = KerHn+1ét (k(R),µ⊗nl ) −→ ⊕q∈(SpecRg )(1)Hnét(κ(q),µ⊗(n−1)l ) .If we haveHn+1Zar (R,Z/l (n)) = Hn+2Zar (R,Z/l (n)) = 0for any regular local ring R which is finite type over a discrete valuation ring and a positiveinteger l which is invertible in R, thenHn+1Zar (Rg ,Z/l (n)) = Hn+2Zar (Rg ,Z/l (n)) = 0by the localization theorem ([6, p. 779, Theorem 3.2]) and we can show thatHn+1ét (Rg ,µ⊗nl ) = Γ(SpecRg ,Rn+1ε∗(µ⊗nl )) = Hn+1ét (Rg ,µ⊗nl )′and Proposition 6 holds. Here ε : (SpecRg )ét → (SpecRg )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 whichis invertible in R. Then the sequence0 −→ Hiét(R,µ⊗nl)−→ Hiét (k(R),µ⊗nl )−→ ⊕p∈(SpecR)(1)Hi−1ét(κ(p),µ⊗(n−1)l)−→ ⊕p∈(SpecR)(2)Hi−2ét(κ(p),µ⊗(n−2)l)−→ 0 (8)is exact for any i ≥ 0.Proof. The exactness of the complex (8) at the first two terms follows from Proposition 5 andProposition 6.We consider the coniveau spectral sequenceHp,q1 =∐x∈(SpecR)(p)Hp+qx(SpecR,µ⊗nl)⇒ Hp+qét (R,µ⊗nl )= Hp+q(cf. [4, §1]). Then we have a filtration0 ⊂ Hp+qp+q ⊂ ·· · ⊂ Hp+q1 ⊂ Hp+q0 = Hp+q ,such thatHp+qp /Hp+qp+1 ' Hp,q∞ .By Theorem 4, it suffices to show thatH1,i−12 = H2,i−22 = 0.By Proposition 5, the morphismHi −→ H0,i∞is injective andHi1 = Hi2 = 0.Hence we haveH1,i−1∞ = H2,i−2∞ = 0.SinceHp,i−p+1r = 0C. R. Mathématique, 2020, 358, n 1, 33-39Makoto Sakagaito 39for p ≥ 3 andH1−r,i+r−2r = 0for r ≥ 2, we haveH1,i−12 = H1,i−1∞ = 0.By the exactness of the complex (8) at the second term, we haveH0,i−12 = H0,i−1∞ = Hi−1andIm(H0,i−12d0,i−12−−−−→ H2,i−22)= 0.Hence we haveH2,i−22 = H2,i−23 .Moreover, sinceH2−r,i+r−3r = 0for r ≥ 3, we haveH2,i−2r+1 =Ker(d2,i−2r )Im(d2−r,i+r−3r )= H2,i−2rfor r ≥ 3. ThereforeH2,i−22 = H2,i−23 = H2,i−2∞ = 0.This completes the proof. References[1] M. Artin, Grothendieck Topologies, Harvard University, 1962.[2] S. Bloch, A. Ogus, “Gersten’s conjecture and the homology of schemes”, Ann. Sci. Éc. Norm. Supér. 7 (1974), p. 181-201.[3] J.-L. Colliot-Thélène, “Quelques problèmes locaux-globaux”, personal notes, 2011.[4] J.-L. Colliot-Thélène, R. T. Hoobler, B. Kahn, “The Bloch–Ogus–Gabber theorem”, in Algebraic K-theory, FieldsInstitute Communications, vol. 16, American Mathematical Society, 1997, p. 31-94.[5] K. Fujiwara, “A proof of the absolute purity conjecture (after Gabber)”, in Algebraic geometry 2000, Azumino (Hotaka),Advanced Studies in Pure Mathematics, vol. 36, Mathematical Society of Japan, 2000, p. 153-183.[6] T. Geisser, “Motivic cohomology over Dedekind rings”, Math. Z. 248 (2004), no. 4, p. 773-794.[7] D. Harbater, J. Hartmann, D. Krashen, “Local-global principles for Galois cohomology”, Comment. Math. Helv. 89(2014), no. 1, p. 215-253.[8] Y. Hu, “A Cohomological Hasse Principle Over Two-dimensional Local Rings”, Int. Math. Res. Not. 2017 (2017), no. 14,p. 4369-4397.[9] J. S. Milne, Étale Cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, 1980.[10] I. A. Panin, “The equicharacteristic case of the Gersten conjecture”, Tr. Mat. Inst. Im. V. A. Steklova 241 (2003), no. 2,p. 169-178.[11] S. Saito, “Arithmetic on two-dimensional local rings”, Invent. Math. 85 (1986), p. 379-414.[12] M. Sakagaito, “On problems about a generalization of the Brauer group”, https://arxiv.org/abs/1511.09232v2, 2016.[13] ——— , “On a generalized Brauer group in mixed characteristic cases”, https://arxiv.org/abs/1710.11449v2,2019.[14] V. Voevodsky, “On motivic cohomology with Z/l-coefficients”, Ann. Math. 174 (2011), no. 1, p. 401-438.C. R. Mathématique, 2020, 358, n 1, 33-39

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

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Comptes RendusMathématiqueMakoto SakagaitoA note on Gersten’s conjecture for étale cohomology overtwo-dimensional henselian regular local ringsVolume 358, issue 1 (2020), p. 33-39Published online: 18 March 2020https://doi.org/10.5802/crmath.9This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orge-ISSN : 1778-3569Comptes RendusMathématique2020, 358, n 1, p. 33-39https://doi.org/10.5802/crmath.9Number Theory, Homological Algebra / Théorie des nombres, Algèbre homologiqueA note on Gersten’s conjecture for étalecohomology over two-dimensional henselianregular local ringsUne note sur la conjecture de Gersten pour lacohomologie étale sur des anneaux locaux réguliershenséliens à deux dimensionsMakoto Sakagaitoaa Indian Institute of Science Education and Research, Mohali.E-mail: makoto@iiserb.ac.in.Abstract. We prove Gersten’s conjecture for étale cohomology over two dimensional henselian regular localrings without assuming equi-characteristic. As an application, we obtain the local-global principle for Galoiscohomology over mixed characteristic two-dimensional henselian local rings.Résumé. Nous montrons la conjecture de Gersten pour la cohomologie étale sur des anneaux locaux régu-liers henséliens sans supposer de caractère équicaractéristique. En application, nous obtenons le principelocal-global pour la cohomologie de Galois sur des anneaux locaux henséliens à deux dimensions de carac-téristique mixte.Manuscript received 30th April 2019, revised 17th September 2019 and 24th November 2019, accepted 17th De-cember 2019.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/34 Makoto Sakagaito1. IntroductionLet R be an equi-characteristic regular local ring, k(R) the field of fractions of R, l a positiveinteger which is invertible in R and µl the étale sheaf of l -th roots of unity. Then the sequenceof étale cohomology groups0 −→ Hn+1ét(R,µ⊗nl)−→Hn+1ét (k(R),µ⊗nl ) (1)−→ ⊕p∈SpecRht(p)=1Hnét(κ(p),µ⊗(n−1)l)−→ ⊕p∈SpecRht(p)=2Hn−1ét(κ(p),µ⊗(n−2)l)−→ ·· ·is exact by Bloch–Ogus ([2]) and Panin ([10]). Here κ(p) is the residue field of p ∈ SpecR.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 thefield of fractions of a henselian excellent discrete valuation ring A.(b) (local case) The function field of a two-dimensional henselian excellent normal localdomain 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 mapHn+1ét(K ,µ⊗nl)−→ ∏v∈ΩKHn+1ét(Kv ,µ⊗nl)(2)injective ?Here ΩK is the set of normalized discrete valuations on K and Kv is the correspondinghenselization of K for each v ∈ΩK .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, Theo-rem 2.5]) proved that the local-global map (2) is injective in both the semi-global case and thelocal case by an alternative method.If the sequence (1) is exact (at the first two terms) in the case where R is a mixed characteristictwo-dimensional excellent henselian local ring, then the local-global map (2) is injective evenwithout 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) discretevaluation ring, the sequence (1) is exact (cf. [6, Theorem 1.2 and Theorem 3.2b)]).In this paper, we show the following result:Theorem 1 (Theorem 9). Let R be a mixed characteristic two-dimensional excellent henselianlocal ring and l a positive integer which is invertible in R. Then Gersten’s conjecture for étalecohomology with µ⊗nl coefficients holds over SpecR. 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 thefollowing result as an application of Theorem 1 :Theorem 2. With notations as above, assume that A is mixed characteristic and l is a positiveinteger which is invertible in A.In both the semi-global case and the local case, the local-global principle for the Galois coho-mology group Hn+1(K ,µ⊗nl ) 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]).C. R. Mathématique, 2020, 358, n 1, 33-39Makoto Sakagaito 351.1. NotationsFor a scheme X , X (i ) is the set of points of codimension i , k(X ) is the ring of rational functions onX and κ(p) is the residue field of p ∈ X . If X = SpecR, k(SpecR) is abbreviated as k(R). The symbolµl 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, Kthe function field of A and l a positive integer which is invertible in A. Then the homomorphismHiét(A,µ⊗nl ) −→ Hiét(K ,µ⊗nl )is injective for any i ≥ 0.Theorem 4 (The absolute purity theorem [5, p. 159, Theorem 2.1.1]). Let Yi,→ X be a closedimmersion of noetherian regular schemes of pure codimension c. Let n be an integer which isinvertible on X , and let Λ= Z/n. Then the cycle class (cf. [5, 1.1]) give an isomorphismΛY∼−→ Ri !Λ(c)[2c]in D+(Yét,Λ). Here D+(Yét,Λ) is the derived category of complexes bounded below of étale sheavesof Λ-modules on Y .In this section, we use Theorem 4 in the case where dim X ≤ 2. In this case, Theorem 4 wasproved 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 functionfield of R. Let l be a positive integer such that l ∉m. Then the homomorphismHiét(SpecR,µ⊗nl)−→ Hiét (SpecK ,µ⊗nl ) (3)is injective for any i ≥ 0.Proof. We prove the statement by induction on dim(R). Let R be a discrete valuation ring (whichdoes 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\m2 and p= (a). Then R/p is ahenselian regular local ring of dimension d andk(R/p) = Rp/pRpwhere k(R/p) is the function field of R/p.Therefore the diagramHiét(SpecR,µ⊗nl) //Hiét(SpecRp,µ⊗nl)Hiét(SpecR/p,µ⊗nl) // Hiét (Speck(R/p),µ⊗nl )(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 inductionhypothesis. Hence the homomorphismHiét(SpecR,µ⊗nl)−→ Hiét (SpecRp,µ⊗nl )C. R. Mathématique, 2020, 358, n 1, 33-3936 Makoto Sakagaitois injective. Moreover the homomorphismHiét(SpecRp,µ⊗nl)−→ Hiét (SpecK ,µ⊗nl )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 whichis invertible in R. Suppose that dim(R) = 2. Then the sequenceHiét(R,µ⊗nl ) −→ Hiét(k(R),µ⊗nl )(∗)−−→ ⊕p∈(SpecR)(1)Hi−1ét (κ(p),µ⊗(n−1)l )is exact for any i ≥ 0.Proof. Let A be a Dedekind ring, q a maximal ideal of A. ThenHi+1q ((Spec A)ét,µ⊗nl ) = Hi−1ét (κ(q),µ⊗(n−1)l )by Theorem 4. Hence the sequenceHiét(A,µ⊗nl ) −→ Hiét(U ,µ⊗nl ) −→⊕q∈Z (1)Hi−1ét (κ(q),µ⊗(n−1)l )is exact where Z is a closed subscheme of Spec A and U = SpecR \ Z . Sincelim→UHiét(U ,µ⊗nl ) = Hiét(k(A),µ⊗nl )by [9, pp. 88–89, III, Lemma 1.16], the sequenceHiét(A,µ⊗nl ) −→ Hiét(k(A),µ⊗nl ) −→⊕q∈(Spec A)(1)Hi−1ét (κ(q),µ⊗(n−1)l ) (5)is exact.Let m be the maximal ideal of R. Let g ∈m\m2, p= (g ) and Z = SpecR/p. Then R/p is a regularlocal ring and we haveHi+1Z ((SpecR)ét,µ⊗nl ) = Hi−1ét (R/p,µ⊗(n−1)l )by Theorem 4.We consider the commutative diagramHiét(R,µ⊗nl )//Hiét(Rg ,µ⊗nl )//Hi−1ét (R/p,µ⊗(n−1)l )′// 00 // Ker(∗) // Hiét(Rg ,µ⊗nl )′ // Hi−1ét (k(R/p),µ⊗(n−1)l )(6)whereHi−1ét (R/p,µ⊗(n−1)l )′ = Im(Hiét(Rg ,µ⊗nl ) −→ Hi−1ét (R/p,µ⊗(n−1)l ))andHiét(Rg ,µ⊗nl )′ = KerHiét(k(Rg ),µ⊗nl ) −→ ⊕q∈(SpecRg )(1)Hi−1ét (κ(q)),µ⊗(n−1)l ) .Then the rows in the diagram (6) are exact by Theorem 4. Since Rg is a Dedekind domain, themiddle map in the diagram (6) is surjective by (5). Moreover, sinceHi−1ét (R/p,µ⊗(n−1)l )′ ⊂ Hi−1ét (R/p,µ⊗(n−1)l )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. C. R. Mathématique, 2020, 358, n 1, 33-39Makoto Sakagaito 37Corollary 7. Let R be the henselization of a regular local ring which is essentially of finite type overa mixed characteristic discrete valuation ring. Suppose that dim(R) = 2. ThenHn+1Zar (R,Z/l (n)) = 0for a positive integer l which is invertible in R. Here Z(n) is Bloch’s cycle complex and Z/l (n) =Z(n)⊗Z/l (cf. [6, p. 779]).Proof. Let m be the maximal ideal of R. Let g ∈m\m2 and p= (g ). Then the homomorphismHn+1ét (R,µ⊗nl ) −→ Hn+1ét (Rg ,µ⊗nl )is injective by Proposition 5. Hence the homomorphismHnét(Rg ,µ⊗nl ) −→ Hn−1ét (R/p,µ⊗n−1l )is surjective by Theorem 4. Therefore the homomorphismHnZar(Rg ,Z/l (n)) −→ Hn−1Zar (R/p,Z/l (n −1))is surjective by [6, p. 774, Theorem 1.2] and [14]. Moreover the homomorphismHn+1Zar (R,Z/l (n)) −→ Hn+1Zar (Rg ,Z/l (n))is injective by the localization theorem [6, p. 779, Theorem 3.2]. We consider the commutativediagramHn+1Zar (Rg ,Z/l (n)) //Hn+1ét (Rg ,Z/l (n))Hn+1Zar (k(Rg ),Z/l (n)) // Hn+1ét (k(Rg ),Z/l (n)).(7)Then the upper map in the commutative diagram (7) is injective by the Beilinson–Lichenbaumconjecture ([6, p. 774, Theorem 1.2], [14]) and the right map in the commutative diagram (7) is in-jective by the commutative diagram (6) in the proof of Proposition 6. Hence the homomorphismHn+1Zar (Rg ,Z/l (n)) −→ Hn+1Zar (k(Rg ),Z/l (n))is injective and the homomorphismHn+1Zar (R,Z/l (n)) −→ Hn+1Zar (k(Rg ),Z/l (n))is also injective. SinceHn+1Zar (k(Rg ),Z/l (n)) = 0,we haveHn+1Zar (R,Z/l (n)) = 0.This completes the proof. Remark 8.(i) If R is a local ring of a smooth algebra over a discrete valuation ring, thenHiZar(R,Z/m(n)) = 0for i > n and any positive integer m (cf. [6, p. 786, Corollary 4.4]).(ii) If we haveHn+1Zar (R,Z/l (n)) = 0for any regular local ring R which is finite type over a discrete valuation ring and a positiveinteger l which is invertible in R, we can show that the homomorphismHn+1ét (R,µ⊗nl ) −→ Hn+1ét (k(R),µ⊗nl )is injective by a similar argument as in the proof of [13, Theorem 4.2].C. R. Mathématique, 2020, 358, n 1, 33-3938 Makoto Sakagaito(iii) The reason why we assume dim(R) = 2 in Propositin 6 and Theorem 9 is that we have toshow that the middle map in the diagram (6), i.e., the homomorphismHn+1ét (Rg ,µ⊗nl ) −→ Hn+1ét (Rg ,µ⊗nl )′is surjective for an element g of m\m2. Here m is the maximal ideal of R andHn+1ét (Rg ,µ⊗nl )′ = KerHn+1ét (k(R),µ⊗nl ) −→ ⊕q∈(SpecRg )(1)Hnét(κ(q),µ⊗(n−1)l ) .If we haveHn+1Zar (R,Z/l (n)) = Hn+2Zar (R,Z/l (n)) = 0for any regular local ring R which is finite type over a discrete valuation ring and a positiveinteger l which is invertible in R, thenHn+1Zar (Rg ,Z/l (n)) = Hn+2Zar (Rg ,Z/l (n)) = 0by the localization theorem ([6, p. 779, Theorem 3.2]) and we can show thatHn+1ét (Rg ,µ⊗nl ) = Γ(SpecRg ,Rn+1ε∗(µ⊗nl )) = Hn+1ét (Rg ,µ⊗nl )′and Proposition 6 holds. Here ε : (SpecRg )ét → (SpecRg )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 whichis invertible in R. Then the sequence0 −→ Hiét(R,µ⊗nl)−→ Hiét (k(R),µ⊗nl )−→ ⊕p∈(SpecR)(1)Hi−1ét(κ(p),µ⊗(n−1)l)−→ ⊕p∈(SpecR)(2)Hi−2ét(κ(p),µ⊗(n−2)l)−→ 0 (8)is exact for any i ≥ 0.Proof. The exactness of the complex (8) at the first two terms follows from Proposition 5 andProposition 6.We consider the coniveau spectral sequenceHp,q1 =∐x∈(SpecR)(p)Hp+qx(SpecR,µ⊗nl)⇒ Hp+qét (R,µ⊗nl )= Hp+q(cf. [4, §1]). Then we have a filtration0 ⊂ Hp+qp+q ⊂ ·· · ⊂ Hp+q1 ⊂ Hp+q0 = Hp+q ,such thatHp+qp /Hp+qp+1 ' Hp,q∞ .By Theorem 4, it suffices to show thatH1,i−12 = H2,i−22 = 0.By Proposition 5, the morphismHi −→ H0,i∞is injective andHi1 = Hi2 = 0.Hence we haveH1,i−1∞ = H2,i−2∞ = 0.SinceHp,i−p+1r = 0C. R. Mathématique, 2020, 358, n 1, 33-39Makoto Sakagaito 39for p ≥ 3 andH1−r,i+r−2r = 0for r ≥ 2, we haveH1,i−12 = H1,i−1∞ = 0.By the exactness of the complex (8) at the second term, we haveH0,i−12 = H0,i−1∞ = Hi−1andIm(H0,i−12d0,i−12−−−−→ H2,i−22)= 0.Hence we haveH2,i−22 = H2,i−23 .Moreover, sinceH2−r,i+r−3r = 0for r ≥ 3, we haveH2,i−2r+1 =Ker(d2,i−2r )Im(d2−r,i+r−3r )= H2,i−2rfor r ≥ 3. ThereforeH2,i−22 = H2,i−23 = H2,i−2∞ = 0.This completes the proof. References[1] M. Artin, Grothendieck Topologies, Harvard University, 1962.[2] S. Bloch, A. Ogus, “Gersten’s conjecture and the homology of schemes”, Ann. Sci. Éc. Norm. Supér. 7 (1974), p. 181-201.[3] J.-L. Colliot-Thélène, “Quelques problèmes locaux-globaux”, personal notes, 2011.[4] J.-L. Colliot-Thélène, R. T. Hoobler, B. Kahn, “The Bloch–Ogus–Gabber theorem”, in Algebraic K-theory, FieldsInstitute Communications, vol. 16, American Mathematical Society, 1997, p. 31-94.[5] K. Fujiwara, “A proof of the absolute purity conjecture (after Gabber)”, in Algebraic geometry 2000, Azumino (Hotaka),Advanced Studies in Pure Mathematics, vol. 36, Mathematical Society of Japan, 2000, p. 153-183.[6] T. Geisser, “Motivic cohomology over Dedekind rings”, Math. Z. 248 (2004), no. 4, p. 773-794.[7] D. Harbater, J. Hartmann, D. Krashen, “Local-global principles for Galois cohomology”, Comment. Math. Helv. 89(2014), no. 1, p. 215-253.[8] Y. Hu, “A Cohomological Hasse Principle Over Two-dimensional Local Rings”, Int. Math. Res. Not. 2017 (2017), no. 14,p. 4369-4397.[9] J. S. Milne, Étale Cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, 1980.[10] I. A. Panin, “The equicharacteristic case of the Gersten conjecture”, Tr. Mat. Inst. Im. V. A. Steklova 241 (2003), no. 2,p. 169-178.[11] S. Saito, “Arithmetic on two-dimensional local rings”, Invent. Math. 85 (1986), p. 379-414.[12] M. Sakagaito, “On problems about a generalization of the Brauer group”, https://arxiv.org/abs/1511.09232v2, 2016.[13] ——— , “On a generalized Brauer group in mixed characteristic cases”, https://arxiv.org/abs/1710.11449v2,2019.[14] V. Voevodsky, “On motivic cohomology with Z/l-coefficients”, Ann. Math. 174 (2011), no. 1, p. 401-438.C. R. Mathématique, 2020, 358, n 1, 33-39

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

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

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