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

Abstract

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.

arXiv comment12 pages. Final version accepted at Comptes Rendus. Mathématique

References in corpus (11)

• A universal characterization of higher algebraic K-theory
• Algebraic K-theory and descent for blow-ups
• On the K-theory of pullbacks
• Hyperdescent and étale K-theory
• Relative Riemann-Zariski spaces
• Excision in algebraic K-theory revisited
• Towards a non-archimedean analytic analog of the Bass-Quillen conjecture
• The tame site of a scheme
• Towards Vorst's conjecture in positive characteristic
• K-theory of admissible Zariski-Riemann spaces
• $K$-theory of rings of continuous functions