Max-Albert Knus, Jean-Pierre Tignol: Thin Severi--Brauer varieties,

Submission: 2009, Dec 17

Severi--Brauer varieties are twisted forms of projective spaces (in the sense of Galois cohomology) and are associated in a functorial way to central simple algebras. Similarly quadrics are related to algebras with involution. Since thin projective spaces are finite sets, thin Severi--Brauer varieties are finite sets endowed with a Galois action; they are associated to étale algebras. Similarly, thin quadrics are étale algebras with involution. We discuss embeddings of thin Severi--Brauer varieties and thin quadrics in Severi--Brauer varieties and quadrics as geometric analogues of embeddings of étale algebras into central simple algebras (with or without involution), and consider the geometric counterpart of the Clifford algebra construction.

2000 Mathematics Subject Classification: 16H05

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