GROTHENDIECK—SERRE CONJECTURE FOR ADJOINT GROUPS OF TYPES E_{6} AND E_{7}

I. PANIN, V. PETROV, AND A. STAVROVA

Abstract. Assume that R is a semi-local regular ring containing an infinite perfect field, or that R is a semi-local ring of several points on a smooth scheme over an infinite field. Let K be the field of fractions of R. Let H be a strongly inner adjoint simple algebraic group of type E_{6} or E_{7} over R. We prove that the kernel of the map

H_{ét}^{1}(R, H) → H_{ét}^{1}(K, H) induced by the inclusion of R into K is trivial. This continues the recent series of papers [PaSV], [Pa] on the Grothendieck—Serre conjecture [Gr, Rem. 1.11].

In what follows we use the notation and terminology of [PS]. Our numbering of vertices of Dynkin diagrams follows [B].

Lemma 1. Let R be a regular semi-local domain and let K be the the field of fractions of R. Let H be a simple group scheme of inner type over R such that H ×_{Spec R} Spec K is of strongly inner type. Then H is of strongly inner type.

Proof. Let A be a Tits algebra of H. By the condition, the class [A ⊗_{R} K] is trivial in Br(K). By [Gr, Corollaire 1.10] [A] is trivial in Br(R). So all Tits algebras of H are trivial, and therefore H is strongly inner. □

Lemma 2. Let R be a semi-local domain, and let H be a strongly inner simply connected simple group scheme of type E_{6} (resp., E_{7}) over R. There exists an inner simply connected simple group scheme G of type E_{7} (resp., E_{8}) over R, together with a maximal parabolic subgroup P of type {1, 2, 3, 4, 5, 6} (resp., {1, 2, 3, 4, 5, 6, 7}), such that H is isomorphic to the derived subgroup of a Levi subgroup of P. Such a group scheme G is unique up to an isomorphism.

Proof. Let H_{0} be a split simply-connected algebraic group over R of the same type as H, and let G_{0} be a split simply-connected algebraic group over R of type E_{7} (respectively, E_{8}) if H is of type E_{6} (respectively, E_{7}). Let P_{0} be a standard maximal parabolic subgroup of G_{0} corresponding to the 7th (respectively, the 8th) vertex of the Dynkin diagram of G_{0}. Then H_{0} is isomorphic to the derived subgroup of a standard Levi subgroup L_{0} of P_{0}. By [PS, Th. 2 (2)] for any strongly inner form H of H_{0} there exist an inner form G of G_{0}, a parabolic subgroup P of G of the same type as P_{0} in G_{0}, and a Levi subgroup L of P, such that H is isomorphic to the derived subgroup of L. By [PS, Th. 2 (3)] if Pic(R) = 0, which is our case, such a G is unique up to an isomorphism. □

Theorem 1. Let R be a semi-local domain. Assume moreover that R is regular and contains a infinite perfect field k, or that R is a semi-local ring of several points on a k-smooth scheme over an infinite field k. Let K be the field of fractions of R. Let H be an adjoint strongly inner simple group scheme of type E_{6} or E_{7} over R. Then the map

H_{ét}^{1}(R, H) → H_{ét}^{1}(K, H)

induced by the inclusion of R into K has trivial kernel.

V. Petrov is partially supported by PIMS fellowship and RFBR 08-01-00756; V. Petrov and A. Stavrova are supported by RFBR 09-01-00878; I.Panin is supported by the joint DFG–RFBR project 09-01-91333-NNIO-a.

Proof. Let H_{0} be a split simply-connected algebraic group over R of the same type as H. The elements of H_{ét}^{1}(R, H) parametrize isomorphism classes of inner forms of H_{0}. Let H_{1} and H_{2} be two inner forms of H_{0}, and assume that

H_{1} ×_{Spec R} Spec K ≅ H_{2} ×_{Spec R} Spec K.

Assume that H_{1} is of strongly inner type. Then H_{2} is also of strongly inner type by Lemma 1. Now by Lemma 2 there exist two simply connected R-group schemes G_{i}, i = 1, 2, of inner type E_{7} (resp., E_{8}), if H is of type E_{6} (resp., E_{7}), together with parabolic subgroups P_{i}, i = 1, 2, of type {1, 2, 3, 4, 5, 6} (resp., {1, 2, 3, 4, 5, 6, 7}), such that H_{i}, i = 1, 2, is isomorphic to the derived subgroup of a Levi subgroup L_{i} of P_{i}. Since H_{1}×_{Spec R}Spec K ≅ H_{2}×_{Spec R}Spec K, the uniqueness part of Lemma 2 applied to R = K implies that

G_{1} ×_{Spec R} Spec K ≅ G_{2} ×_{Spec R} Spec K.

Since the groups G_{i}, i = 1, 2, are isotropic, and are inner forms of each other, by [Pa, Th. 1.0.1, Th. 1.0.2] they are isomorphic over R. We can assume G_{1} = G_{2}. Since R is semi-local, by [SGA, Exp. XXVI Cor. 5.5 (iv)] there exists g ∈ G_{1}(R) such that gP_{1}g^{−1} = P_{2} and gL_{1}g^{−1} = L_{2}. Then the derived subgroups of L_{1} and L_{2} are isomorphic, that is, H_{1} ≅ H_{2} over R. □

Remark. Actually, the proof is also valid for adjoint (semi)simple algebraic groups of classical type.

References

[B] N. Bourbaki, Groupes et algèbres de Lie. Chapitres 4, 5 et 6, Masson, Paris, 1981.

[Gr] A. Grothendieck, Le groupe de Brauer II, Sém. Bourbaki 297 (1965/66).

[SGA] M. Demazure, A. Grothendieck, Schémas en groupes, Lecture Notes in Mathematics, Vol. 151–153, Springer-Verlag, Berlin-Heidelberg-New York, 1970.

[PaSV] I. Panin, A. Stavrova, N. Vavilov, On Grothendieck—Serre’s conjecture concerning principal G-bundles over reductive group schemes:I, Preprint (2009), http://www.math.uiuc.edu/K-theory/

[Pa] I. Panin, On Grothendieck—Serre’s conjecture concerning principal G-bundles over reductive group schemes:II, Preprint (2009), http://www.math.uiuc.edu/K-theory/

[PS] V. Petrov, A. Stavrova, Tits indices over semilocal rings, Preprint (2008), available from http://www.arxiv.org/abs/0807.2140