Albert algebras over locally ringed spaces

The main results are as follows. We first carry over the Tits process to the setting of Jordan algebras over locally ringed spaces, in analogy to the Cayley-Dickson doubling due to H. P. Petersson. This yields a generalization of the original Tits process even in the case of rings, i.e., affine schemes. Then it is shown that an Albert algebra J over a scheme (X, O_X) is obtained by the Tits process under one of the following assumptions: Either J contains a subalgebra of the form A⁺ where A is an Azumaya algebra of rank 9 over O_X, or it contains a subalgebra H(B,*) where B is an Azumaya algebra of rank 9 with involution * of the second kind over a quadratic extension of O_X. As an application, we obtain explicit examples of exceptional Jordan algebras over rings.

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