A_0(SB_r(A)) is (d, n/d)-torsion with d = gcd(r,n) and vanishes over local or global fields, extending to stably birational Severi-Brauer flag varieties.
Local-global principles.https://virtualmath1.stanford.edu/ ~conrad/248APage/handouts/ localglobal.pdf, n.d
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Zero cycles on Severi--Brauer flag varieties
A_0(SB_r(A)) is (d, n/d)-torsion with d = gcd(r,n) and vanishes over local or global fields, extending to stably birational Severi-Brauer flag varieties.