Cohomological invariants and R-triviality of adjoint classical groups

Abstract

Abstract. Using a cohomological obstruction, we construct examples of absolutely simple adjoint classical groups of type 2 An with n ≡ 3 mod 4, Cn or 1 Dn with n ≡ 0 mod 4, which are not R-trivial hence not stably rational.

Finiteness of R-equivalence groups of some adjoint classical groups of type D32

“…By assumption, the kernel of the map (∂ x ) x∈X (1) : Im(ϕ) → x∈X (1) H 2 κ(x), μ 2 is finite. By the previous lemma, its image is finite as well, so we are done by Remark 3.…”

“…Let x ∈ X (1) , and assume first that Proof. By assumption, the kernel of the map (∂ x ) x∈X (1) : Im(ϕ) → x∈X (1) H 2 κ(x), μ 2 is finite.…”

“…Let X be a smooth proper irreducible variety defined over k. We denote by X (1) the set of points of codimension 1 in X. The ring O X,x is then a discrete valuation ring.…”

“…It is well known that the set of poles of α is finite. The unramified cohomology group H n nr (k(X), μ 2 ) is the subgroup of H n (k(X), μ 2 ) consisting of classes which are unramified at every x ∈ X (1) . It is a birational invariant of X.…”

“…[4,Proposition 14]) and by Gille for any reductive group G defined over a global field in [5]. Lemma II.1.1(c) of [5] immediately implies that this question has a positive answer if F is a rational extension of a global field k and G is defined over k. Various examples of classical adjoint groups which are not R-trivial were constructed in [1] or [6], [11]. Throughout this paper, we will assume that F is a field of characteristic different from 2 and we will focus on absolutely simple adjoint groups of type 2 D 3 .…”