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

TL;DR

This paper computes the group of $K_1$-zero-cycles on a specific algebraic variety related to algebra of index 4, linking it to $K_1(A)$ and square roots of reduced norms, using advanced algebraic tools.

Contribution

It provides an explicit description of the $K_1$-zero-cycle group for the second generalized Severi-Brauer variety of an algebra with index 4, connecting it to algebraic groups and involution varieties.

Findings

The group of $K_1$-zero-cycles is characterized by elements of $K_1(A)$ and their reduced norm square roots.

Explicit algebraic group descriptions are provided for the computation.

The approach translates the problem to involution varieties and uses Clifford and spin groups.

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.