On relative and bi-relative algebraic 𝐾-theory of rings of finite characteristic

We consider unital associative rings in which a fixed prime number p p is nilpotent. It was proved long ago by Weibel that for such rings, the relative K K -groups associated with a nilpotent extension and the bi-relative K K -groups associated with a pull-back square are p p -primary torsion groups. However, the question of whether these groups can contain a p p -divisible torsion subgroup has remained an open and intractable problem. In this paper, we answer this question in the negative. In effect, we prove the stronger statement that the groups in question are always p p -primary torsion groups of bounded exponent.