The group of $K_1$-zero-cycles on the second generalized Severi-Brauer variety of an algebra of index 4

Abstract

In this manuscript, it is shown that the group of $K_1$-zero-cycles on the second generalized Severi-Brauer variety of an algebra $A$ of index 4 is given by elements of the group $K_1(A)$ together with a square-root of their reduced norm. Utilizing results of Krashen concerning exceptional isomorphisms, we 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.