logo CRAS
Comptes Rendus. Mathématique
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]
Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 33-39.

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.

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.

Reçu le : 2019-04-29
Révisé le : 2019-09-16
Accepté le : 2019-12-16
Publié le : 2020-03-18
DOI : https://doi.org/10.5802/crmath.9
@article{CRMATH_2020__358_1_33_0,
     author = {Makoto Sakagaito},
     title = {A note on Gersten's conjecture for \'etale cohomology over two-dimensional henselian regular local rings},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {33--39},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {1},
     year = {2020},
     doi = {10.5802/crmath.9},
     language = {en},
     url = {comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_1_33_0/}
}
Makoto Sakagaito. A note on Gersten’s conjecture for étale cohomology over two-dimensional henselian regular local rings. Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 33-39. doi : 10.5802/crmath.9. https://comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_1_33_0/

[1] Michael Artin Grothendieck Topologies, Harvard University, 1962 | Zbl 0208.48701

[2] Spencer Bloch; Arthur Ogus Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Supér., Volume 7 (1974), pp. 181-201 | Article | MR 412191 | Zbl 0307.14008

[3] Jean-Louis Colliot-Thélène Quelques problèmes locaux-globaux (2011) (personal notes)

[4] Jean-Louis Colliot-Thélène; Raymond T. Hoobler; Bruno Kahn The Bloch–Ogus–Gabber theorem, Algebraic K-theory (Fields Institute Communications) Volume 16, American Mathematical Society, 1997, pp. 31-94 | MR 1466971 | Zbl 0911.14004

[5] Kazuhiro Fujiwara A proof of the absolute purity conjecture (after Gabber), Algebraic geometry 2000, Azumino (Hotaka) (Advanced Studies in Pure Mathematics) Volume 36, Mathematical Society of Japan, 2000, pp. 153-183 | MR 1971516 | Zbl 1059.14026

[6] Thomas Geisser Motivic cohomology over Dedekind rings, Math. Z., Volume 248 (2004) no. 4, pp. 773-794 | Article | Zbl 1062.14025

[7] David Harbater; Julia Hartmann; Daniel Krashen Local-global principles for Galois cohomology, Comment. Math. Helv., Volume 89 (2014) no. 1, pp. 215-253 | Article | MR 3177913 | Zbl 1332.11046

[8] Yong Hu A Cohomological Hasse Principle Over Two-dimensional Local Rings, Int. Math. Res. Not., Volume 2017 (2017) no. 14, pp. 4369-4397 | MR 3674174 | Zbl 1405.11042

[9] James S. Milne Étale Cohomology, Princeton Mathematical Series, Volume 33, Princeton University Press, 1980 | MR 559531 | Zbl 0433.14012

[10] Ivan A. Panin The equicharacteristic case of the Gersten conjecture, Tr. Mat. Inst. Im. V. A. Steklova, Volume 241 (2003) no. 2, pp. 169-178 | MR 2024050 | Zbl 1115.19300

[11] Shuji Saito Arithmetic on two-dimensional local rings, Invent. Math., Volume 85 (1986), pp. 379-414 | Article | MR 846934 | Zbl 0609.13003

[12] Makoto Sakagaito On problems about a generalization of the Brauer group (2016) (https://arxiv.org/abs/1511.09232v2)

[13] Makoto Sakagaito On a generalized Brauer group in mixed characteristic cases (2019) (https://arxiv.org/abs/1710.11449v2)

[14] Vladimir Voevodsky On motivic cohomology with Z/l-coefficients, Ann. Math., Volume 174 (2011) no. 1, pp. 401-438 | MR 2811603 | Zbl 1236.14026