Mayer-Vietoris sequence in algebraic K-theory ($K_{1}A\to K_1{A}_1\oplus K_1{A}_2\to K_1{A}_0\to K_{0}A\to K_0{A}_1\oplus K_0{A}_2\to K_0{A}_0$)

The diagonal maps are extension of scalars; the maps to $A_0$ are differences. The connecting map sends $g$ to the difference of its patched free module and the corresponding free $A$-module. Elementary lifting and common free stabilization prove exactness at every internal term. The displayed fragment does not assert injectivity at its first term or surjectivity at its last.