Zero cycles on Severi--Brauer flag varieties

Let $A$ be a central simple algebra over a field $F$ with index $n$ and let ${\mathrm{S}\mathrm{B}}_r(A)$ denote the $r$-th generalized Severi–Brauer variety associated with $A$. We prove that the Chow group of zero cycles of degree zero ${\mathrm{A}}_0({\mathrm{S}\mathrm{B}}_r(A))$ is $(d,n\mathrm{/}d)$-torsion where $d=(r,n)$. Our approach reduces the general case to division algebras of prime power index and yields several new instances in which ${\mathrm{A}}_0$ is trivial, together with sharper torsion bounds in general.\ We also show that if $F$ is a local or global field, then ${\mathrm{A}}_0({\mathrm{S}\mathrm{B}}_r(A))=0$. Since Severi–Brauer flag varieties are stably birational to generalized Severi–Brauer varieties, these results extend to them, yielding corresponding torsion bounds and vanishing results for ${\mathrm{A}}_0(X)$, where $X$ is stably birational to ${\mathrm{S}\mathrm{B}}_r(A)$.