# Unramified cohomology of classifying varieties for exceptional simply connected groups

@article{Garibaldi2005UnramifiedCO, title={Unramified cohomology of classifying varieties for exceptional simply connected groups}, author={Ryan Skip Garibaldi}, journal={Transactions of the American Mathematical Society}, year={2005}, volume={358}, pages={359-371} }

Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in (Q/Z)'(2) over an arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over all fields. These computations provide another family of examples of simple simply connected groups G such that BG is not stably rational.

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#### References

SHOWING 1-10 OF 38 REFERENCES

UNRAMIFIED COHOMOLOGY OF CLASSIFYING VARIETIES FOR CLASSICAL SIMPLY CONNECTED GROUPS

- Mathematics
- 2002

Abstract Let F be a field and G⊂ SL n,F an algebraic closed subgroup of SL n,F . Denote by BG the factor variety SL n /G . The stable F-birational type of BG is independent on the choice of an… Expand

Invariants of quasi-trivial tori and the Rost invariant

- Mathematics
- 2003

For any absolutely simple, simply connected linear algebraic group G over a field F, Rost has defined invariants for torsors under G with values in the Galois cohomology group H^3(F,Q/Z(2)). The aim… Expand

K-cohomology of Severi-Brauer Varieties and the norm residue homomorphism

- Mathematics
- 1983

The basic purpose of this paper is to prove bijectivity of the norm residue homomorphism for any field of characteristic prime to . In particular, if , then any central simple algebra of exponent is… Expand

Isotropic Trialitarian Algebraic Groups

- Mathematics
- 1998

Abstract By modifying a construction from Knus et al. , we construct all isotropic algebraic groups of type 3 D 4 and 6 D 4 over an arbitrary field of characteristic ≠ 2. We also provide a nice… Expand

The Rost invariant has trivial kernel for quasi-split groups of low rank

- Mathematics
- 2000

Abstract. For G a simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map $ R_G : H^1(F, G) \rightarrow H^3(F, \mathbb{Q} / \mathbb{Z}(2)) $.… Expand

Structurable Algebras and Groups of Type E6 and E7

- Mathematics
- 2001

Abstract It is well known that every group of type F 4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type 1 E 6 with trivial Tits algebras arise as… Expand

Strongly inner anisotropic forms of simple algebraic groups

- Mathematics
- 1990

(the definition of the index of a semisimple algebraic group over a field will be recalled in 1.55). More precisely, we show, in Proposition 2(B) (cf. 3.1), that if, over a given field k, there… Expand

Algèbres simples centrales de degré 5 et E 8

- Mathematics
- Canadian Mathematical Bulletin
- 2002

Abstract As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree 5 on fields containg a fifth root of unity is equivalent to the… Expand

Structurable algebras and groups of type E_6 and E_7

- Mathematics
- 1998

It is well-known that every algebraic group of type F_4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type ^1E_6 with trivial Tits algebras arise as… Expand

The Book of Involutions

- Mathematics
- 1998

This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups. It aims to provide the algebra-theoretic… Expand