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

Douai, Jean-Claude

J. Théor. Nombres Bordx. 21, No. 1, 119-129 (2009).

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.

Reviewer: Dorin-Mihail Popescu (Bucureşti)

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

Keywords:

Regular model; excellent Henselian local ring; simply connected semi-simple group

Citations:

Zbl 1055.14019

References:

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