Purity conjecture for reductive groups

Abstract

The following problem was posed by J.-L. Colliot-Th èléne and J.-J. Sansuc in [1, p. 124, Problem 6.4]. Given a local regular ring R and a reductive group scheme G over R determine whether the functor S → H 44-01 (S, G) satisfies the property of purity for R. In this work, we study this problem in a number of interesting particular cases. Namely, let k be a characteristic zero field, and G be one of the following algebraic groups over k: PGL _n , SL_1,A , O(q), SO(q), Spin(q), SL _n /μ _d where d divides n (here, A is a central simple k-algebra). In this paper we prove that the functor R → H ¹_ét (R, G) satisfies the property of purity for the group G and a regular local ring containing the field. In view of this result, it would appear reasonable to suggest that the aforementioned functor possesses the property of purity for an arbitrary connected reductive group G over a zero characteristic field k and an arbitrary regular local ring containing the field k. For groups of the types G ₂ and F ₄ with a trivial g₃ invariant, this conjecture has been proved in [2] and [3]. The problem and conjecture formulated above appear to be an extension of the known conjectures proposed by A. Grothendieck and J.-P. Serre (see [5, Remark 3, pp. 26–27], [6, Remark 1.11.a], and [14, Remark on p. 31]).