Isotopes of octonion algebras, G2-torsors and triality

Albert algebras over rings and related torsors

“…S is straight-forward and analogous to that of [AG,Theorem 4.6]. The same goes for surjectivity whenever E p (S) = ∅, and for naturality.…”

“…Set A = H 3 (C, 1). We need to show that the map H 1 (S, Aut(C))−→H 1 (S, Aut(A)), induced by the subgroup inclusion, has non-trivial kernel for a smooth C-ring S and octonion S-algebra C. This inclusion is the composition of the inclusions i : RT(C) → Aut(A) of Lemma 3.10 (noting that RT(C) = Aut(M C )) and j : Aut(C) → RT(C) defined by t → (t, t, t) (see [AG,3.5]). Thus the map of cohomologies factors as H 1 (S, Aut(C)) j * −→ H 1 (S, RT(C)) i * −→ H 1 (S, AutA) and has non-trivial kernel since by [AG,4.3] (which is a variant of [G1, 3.5]), so does j * , for an appropriate (smooth) choice of S. This completes the proof.…”

“…The lemma above implies that the two RT(C)-torsors Aut(A) ∧ Aut(C) RT(C) and Aut(A) × RT(C)/Aut(C) over Aut(A)/Aut(C), which is representable by an affine scheme by [CTS,6.12], are isomorphic. By [AG,Theorem 4.1], the quotient RT(C)/Aut(C) is isomorphic to the product of two copies of the Euclidean 7-sphere S 7 . Using Lemma 1.4 and arguing as in the proof of Theorem 3.15, it thus suffices to show that for some n, π n (Spin 8 (R)) is not a direct factor in Γ = π n (Aut(A)(R)) × π n (S 7 (R)) 2 .…”

Generically split octonion algebras and 𝔸1-homotopy theory

“…In a different direction, one might try to characterize those octonion algebras with a given associated spinor bundle. As mentioned in the Remark 10 Alsaody and Gille have given precisely such a classification [AG17].…”

“…We would like to thank Philippe Gille for email exchanges on classification results for octonion algebras long ago. We also thank him for sharing an early draft of [AG17]. We would also like to thank Brian Conrad for discussions about G 2 and related homogeneous spaces over Z; in particular, he kindly provided Lemma 2.3.7 and its proof to us.…”