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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

Citations:

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
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