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**Classical groups and the Hasse principle.**
*(English)*
Zbl 0909.20029

Ann. Math. (2) 147, No. 3, 651-693 (1998); Correction 163, No. 1, 381 (2006).

The paper adds some important notes to the solution of the “Hasse principle conjectures” on the structure of the cohomology \(H^1(k,G)\) of the algebraic group \(G\), defined over the field \(k\).

In 1962, Serre conjectured the following: – Let \(k\) be a perfect field of cohomological dimension \(\leq 1\). Let \(G\) be a connected linear algebraic group defined over \(k\). Then \(H^1(k,G)=0\). – Let \(k\) be a perfect field of cohomological dimension \(\leq 2\). Let \(G\) be a semi-simple simply connected linear algebraic group defined over \(k\). Then \(H^1(k,G)=0\).

The first was proved by Steinberg in 1965, the second by the authors in 1995, for groups of classical type. Of course, the structure of \(H^1(k,G)\) is strongly related to the ground field \(k\).

In this paper the authors generalize the conjecture by proving, amongst other things, that – If \(k\) is a perfect field such that the cohomological dimension of \(k(\sqrt{-1})\) is \(\leq 2\), and \(G\) a semi-simple, simply connected group, which is a product of groups of type \(A_n\), \(B_n\), \(C_n\), \(D_n\), the case \(D_4\) being excluded, or \(G\) is of type \(G_2\) or \(F_4\), then the natural map \(H^1(k,G)\to\prod_vH^1(k_v,G)\) is injective, where \(v\) runs over the orderings of \(k\) and \(k_v\) denotes the real closure of \(k\) at \(v\). This was proved, for number fields, by Kneser, Springer, Harder and Chernousov, and restricts, for \(k\) having no real places, to the case \(H^1(k,G)=0\).

Added in 2007: From the text of the correction: Lemma 2.4, on page 654, is not correct as stated. Indeed, \(I^3(k)\) torsion-free does not imply \(I^3(k\sqrt{-1})=0\). Hence, throughout the paper the assumption \(I^3(k)\) torsion-free should be replaced by \(I^3(k\sqrt{-1})=0\).

In 1962, Serre conjectured the following: – Let \(k\) be a perfect field of cohomological dimension \(\leq 1\). Let \(G\) be a connected linear algebraic group defined over \(k\). Then \(H^1(k,G)=0\). – Let \(k\) be a perfect field of cohomological dimension \(\leq 2\). Let \(G\) be a semi-simple simply connected linear algebraic group defined over \(k\). Then \(H^1(k,G)=0\).

The first was proved by Steinberg in 1965, the second by the authors in 1995, for groups of classical type. Of course, the structure of \(H^1(k,G)\) is strongly related to the ground field \(k\).

In this paper the authors generalize the conjecture by proving, amongst other things, that – If \(k\) is a perfect field such that the cohomological dimension of \(k(\sqrt{-1})\) is \(\leq 2\), and \(G\) a semi-simple, simply connected group, which is a product of groups of type \(A_n\), \(B_n\), \(C_n\), \(D_n\), the case \(D_4\) being excluded, or \(G\) is of type \(G_2\) or \(F_4\), then the natural map \(H^1(k,G)\to\prod_vH^1(k_v,G)\) is injective, where \(v\) runs over the orderings of \(k\) and \(k_v\) denotes the real closure of \(k\) at \(v\). This was proved, for number fields, by Kneser, Springer, Harder and Chernousov, and restricts, for \(k\) having no real places, to the case \(H^1(k,G)=0\).

Added in 2007: From the text of the correction: Lemma 2.4, on page 654, is not correct as stated. Indeed, \(I^3(k)\) torsion-free does not imply \(I^3(k\sqrt{-1})=0\). Hence, throughout the paper the assumption \(I^3(k)\) torsion-free should be replaced by \(I^3(k\sqrt{-1})=0\).

Reviewer: Giovanni Falcone (Udine)

### MSC:

20G10 | Cohomology theory for linear algebraic groups |

11E72 | Galois cohomology of linear algebraic groups |

11E57 | Classical groups |

12G05 | Galois cohomology |