## On the non-abelian 2-cohomology of regular models of Henselian local rings. (Sur la 2-cohomologie non abelienne des modèles réguliers des anneaux locaux henséliens.)(French. English summary)Zbl 1181.14016

Let $$(A,m,k)$$ be an excellent Henselian, local domain with $$k$$ either finite, or separable closed. Let $$X\rightarrow \mathrm{Spec } A$$ be a proper morphism with special fiber $$X_0\rightarrow \mathrm{Spec } k$$ of dimension at most one. If $$X$$ is regular and $$L$$ is a $$X_{et}$$-lien that is locally representable by a simply connected semi-simple group then all classes of $$H^2(X_{et},L)$$ are neutral. This completes the results of J.-L. Colliot-Thélène, M. Ojanguren, R. Parimala [Proceedings of the international colloquium on algebra, arithmetic and geometry, Mumbai, India, January 4–12, 2000. Part I and II. New Delhi: Narosa Publishing House, published for the Tata Institute of Fundamental Research, Bombay. Stud. Math., Tata Inst. Fundam. Res. 16, 185–217 (2002; Zbl 1055.14019)]. If $$\dim \;A=2$$, $$k$$ is algebraically closed of equicharacteristic $$0$$ and $$X$$ is regular model of $$A$$ then all classes of $$H^2(K,L)$$, $$K=\mathrm{Frac }A$$ are neutral.

### MSC:

 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14F22 Brauer groups of schemes 14F99 (Co)homology theory in algebraic geometry 13F40 Excellent rings 13J15 Henselian rings

Zbl 1055.14019
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### References:

 [1] J.L. Colliot-Thélène, M. Ojanguren and R. Parimala, Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes. In Algebra, Arithmetic and Geometry, I, II (Mumbai, 2000), Tata Inst. Fund. Res. Stud. Math. 16 (2002), 185-217. MR 1940669. · Zbl 1055.14019 [2] J.L. Colliot-Thélène, P. Gille and R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric fields. Duke Math. J. 121 (2004), 285-341. · Zbl 1129.11014 [3] J.C. Douai, 2-cohomologie galoisienne des groupes semi-simples. Thèse d’Etat, Université de Lille 1, Lille, France, 1976. · Zbl 0328.20036 [4] J.C. Douai, Sur la 2-cohomologie galoisienne de la composante résiduellement neutre des groupes réductifs connexes définis sur les corps locaux. C.R. Acad. Sci. Paris, Série I 342 (2006), 813-818. · Zbl 1101.11014 [5] J. Giraud, Cohomologie non abélienne. Grundlheren Math. Wiss. vol. 179, Springer-Verlag, 1971. · Zbl 0226.14011 [6] J.S. Milne, Etale cohomology. Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, 1980. · Zbl 0433.14012 [7] S.G.A.D., Séminaire de géométrie algébrique 1963-1964. Lecture Notes in Math., 151-153, Springer, 1970.
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