Non-stable $K$-theory for $QB$-rings

Abstract

We study the class of $QB$-rings that satisfy the weak cancellation condition of separativity for finitely generated projective modules. This property turns out to be crucial for proving that all (quasi-)invertible matrices over a $QB$-ring can be diagonalised using row and column operations. The main two consequences of this fact are: (i) The natural map $(\mathrm{GL}_1(R)\to K_1(R)$ is surjective, and (ii) the only obstruction to lift invertible elements from a quotient is of $K$-theoretical nature. We also show that for a reasonably large class of $QB$-rings that includes the prime ones, separativity always holds.