K-cohomology of generalized Severi-Brauer varieties

Abstract

In 1992, Merkurjev and Suslin provided an explicit description of the group of K_1-zero-cycles of the Severi-Brauer variety associated to a central simple algebra A. This description was given in terms of the group K_1(A) and yields a cohomological description of pairs consisting of a maximal subfield of A together with an element of this subfield. In this thesis, we compute the group of K_1-zero-cycles of the second generalized Severi-Brauer variety of a central simple algebra A of index 4 in terms of elements of K_1(A) and their reduced norms. Analogously, this group gives a cohomological description of the quadratic subfields of the degree 4 maximal subfields of the algebra A. To give such a description, we utilize work of Krashen to translate our problem to the computation of cycles on involution varieties. Work of Chernousov and Merkurjev then gives a means of describing such cycles in terms of Clifford and spin groups and corresponding R-equivalence classes. We complete our computation by giving an explicit description of these algebraic groups.