# Albert Algebras over Rings and Related Torsors

## TL;DR

This paper explores the structure of Albert algebras over rings, demonstrating the existence of non-isomorphic isotopes and coordinate algebras, and uses torsors to classify and understand these exceptional Jordan algebras.

## Contribution

It introduces a geometric torsor-based framework to analyze Albert algebras over rings, revealing new non-isomorphic structures and generalizations of classical results over fields.

## Key findings

- Albert algebras over rings admit non-isomorphic isotopes.
- Existence of non-trivial torsors shows Albert algebras do not determine underlying compositions.
- Reduced Albert algebras can have non-isomorphic coordinate algebras with non-isometric quadratic forms.

## 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 $\mathrm F_4$-torsor with total space a group of type $\mathrm E_6$, 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 $\mathrm D_4$-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 non-isomorphic and have non-isometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.