Albert algebras over rings and related torsors

Abstract

We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to non-isomorphic structures.We begin by showing that isotopes of Albert algebras are obtained as twists by a certain F 4 -torsor with total space a group of type E 6 , and using this, that Albert algebras over rings in general admit non-isomorphic isotopes, even in the split case as opposed to the situation over fields. We then consider certain D 4 -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is non-trivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have non-isometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

Groups of type $\mathrm{E}_6$ and $\mathrm{E}_7$ over Rings via Brown Algebras and Related Torsors

“…Proof. The two first statements are immediate from the previous proposition and [Als,Proposition 2.1]. For the third, Remark 3.7 establishes the "if"-direction.…”

R‐triviality of groups of type F4 arising from the first Tits construction

“…Case (3): Here, up to isogeny, M is of the form O + (D, f ) where D is a quaternion algebra and f is a skew-Hermitian form over D of dimension 4 with Witt index 1. From [18], we know that after passing to the function field of the Severi-Brauer variety of D, the anisotropic part of f remains anisotropic, hence the Tits index of M is of the form (2). But this case is impossible by the above consideration.…”

“…Second proof By [2, Theorem 2.5], this kernel parametrizes the isomorphism classes of isotopes of . But from [16, Corollary 60], it follows that the invariants mod 2 and 3 of any isotope of the split Albert algebra are trivial.…”

Albert algebras over Z and other rings

“…We have defined Albert algebras by a sort of descent condition in Definition 7.1. In view of Proposition 10.2 below, one could alternatively define an Albert algebra as a cubic Jordan algebra that is projective of rank 27 as an R-module such that J ⊗ F is a simple Jordan F -algebra for every field F ; this is the definition used in [Pet19] and [Als21], for example. The two notions lead to the same objects.…”

“…A recent article by Alsaody, [Als21], gives several interesting examples about Albert algebras over rings, especially concerning isotopy, compare §12 here. That paper relies on the assertion about Aut(J) already mentioned.…”