Unramified Cohomology of Classifying Varieties for Exceptional Simply Connected Groups

Let BG be a classifying variety for an exceptional simple algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in (Q/Z)′(2) over a nearly arbitrary field F . Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over (nearly) all fields. Let G be an algebraic group over a field F with an embedding ρ : G ↪→ SLn over F . The isomorphism class of the variety X := SLn/ρ(G) depends upon the embedding, but its stable birationality type does not. We call X a classifying space of G. We will compute certain invariants of the stable birationality type of BG, specifically the unramified cohomology defined as follows. Let (Q/Z)′(d) be the module lim −→ ⊗d n for n not divisible by char(F ). For each d > 0, define H d nr(F (X)) to be the intersection of the kernels of the residue homorphisms ∂v : H (F (X), (Q/Z)′(d− 1))→ Hd−1(F (v), (Q/Z)′(d− 2)) as v ranges over the discrete valuations of F (X) over F . If K is a purely transcendental extension of F , then the natural map H nr(F (X)) → H nr(K(X)) is an isomorphism, where the discrete valuation rings for the latter group are those containing K [Mer, 2.3]. That is, the group H nr(F (X)) does not depend on ρ, but only upon the stable birationality type BG of X, so we write H nr(BG) for H d nr(F (X)), or H nr(BFG) in order to emphasize the base field F . The natural homomorphism H(F, (Q/Z)′(d − 1)) → H nr(BFG) is split by the evaluation at the distinguished point of BG, thus, H nr(BFG) = H (F, (Q/Z)′(d− 1))⊕H nr(BFG)norm, where the latter group is the group of normalized classes. (For details about all of this, please see [Mer].) The goal of this paper is to complete the computation of H nr(BG)norm for G simply connected semisimple and F (nearly) arbitrary. The computation of H nr(BG)norm for these groups G is quickly reduced to the case where G is simple simply connected [Mer, §4]. In [Mer], H nr(BG)norm was computed for G simple and classical. We compute it for the remaining cases, where G is exceptional, that is, where G is of type G2, D4, D4, F4, E6, E7, or E8. Date: 16 November 2001. 1991 Mathematics Subject Classification. 20G10 (14M17 14M20 17B25). The author was partially supported by the NSF.