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 nonisomorphic 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 nonisomorphic 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 nontrivial, 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 nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

我们从几何的角度研究了交换环上的特殊 Jordan 代数和相关的特殊群方案，使用适当的torsor 来参数化和解释经典和新结构，并证明在环上，它们会产生非同构结构。

我们首先证明 Albert 代数的同位素是通过某个 $\mathrm F_4$ -torsor 以扭曲的形式获得的，总空间为 $\mathrm E_6$ 类型的组，并且使用这个，环上的 Albert 代数通常承认非同构同位素即使在分裂的情况下，而不是超过领域的情况。然后我们考虑某些 $\mathrm D_4$ -torsors 从简化的 Albert 代数构建，并展示这些如何产生一类由二次形式的组合构建的广义简化的 Albert 代数。表明这个torsor 是非平凡的，我们得出结论，即使在分裂的情况下，Albert 代数也不能唯一地确定基础组成。以类似的方式，我们证明给定的约化 Albert 代数可以接纳两个非同构且具有非等距二次型的坐标代数，这在强烈意义上与 Albert 和 Jacobson 建立的域上的情况相反。