Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory

We show that the algebraic K-theory of semi-valuation rings with stably coherent regular semi-fraction ring satisfies homotopy invariance. Moreover, we show that these rings are regular if their valuation is non-trivial. Thus they yield examples of regular rings which are not homotopy invariant for algebraic K-theory. On the other hand, they are not necessarily coherent, so that they provide a class of possibly non-coherent examples for homotopy invariance of algebraic K-theory. As an application, we show that Temkin’s relative Riemann–Zariski spaces also satisfy homotopy invariance for K-theory under some finiteness assumption.