The following conjecture was made by J.-P. Serre and A. Grothendieck in 1958:
Let \(X\) be a smooth, irreducible variety over an algebraically closed field \(k\); let \(G\) be a connected reductive \(k\)-group. Then every principal homogeneous space over \(G\), with base \(X\) and having a rational section, is locally trivial in the Zariski topology.
This conjecture is solved in the paper under review, with weaker assumptions on \(k\) and \(G\). Namely, it holds when
(i) \(k\) is infinite and perfect, and \(G\) is linear and smooth (theorem 3.2); or when
(ii) \(k\) is infinite, \(G\) is reductive, and every \(k\)-simple component of the derived group of \(G\) is \(k\)-isotropic (theorem 2.1).
More generally, the authors consider the following conjecture of A. Grothendieck [in Dix Exposés Cohomologie Schémas, Adv. Stud. Pure Math. 3, 67–87 (1968; Zbl 0198.25803)]:
Let \(A\) be a regular local ring, with the field of fractions \(K\). Let \(G\) be a reductive group scheme over \(A\). Then the map of pointed sets \(H^ 1(A,G)\to H^ 1(K,G_ K)\) has a trivial kernel.
This conjecture is established in various cases; in theorem 1.1, the authors reduce it to checking that the functor \(X\to H^ 1(X,G)\) commutes with inductive limits (Property P1), satisfies a weak form of homotopic invariance (P2), and satisfies a glueing property (P3). The most difficult point is (P2), for which results of M. S. Raghunathan [Math. Ann. 285, No. 2, 309–332 (1989; Zbl 0672.14007)] are used. This approach can be followed for other functors, e.g. \(K\)- functors; see §4. In §5, an anlogue of Gersten’s conjecture is formulated and studied, by expanding ideas of D. Quillen [in Algebraic \(K\)-theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85–147 (1973; Zbl 0292.18004)].