Comptes Rendus
Research article - Algebra, Algebraic geometry
Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory
Christian Dahlhausen1 
1 Institut für Mathematik, Fakultät für Mathematik und Informatik, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 989-1001

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.

Nous montrons que la K-théorie algébrique des anneaux de semi-valuation avec un anneau de semi-fractions régulier et stablement cohérent satisfait à l’invariance par homotopie. De plus, nous montrons que ces anneaux sont réguliers si leur valuation est non-triviale. Ainsi, ils donnent des exemples d’anneaux réguliers qui ne sont pas invariants par homotopie pour la K-théorie algébrique. D’autre part, ils ne sont pas nécessairement cohérents, de sorte qu’ils fournissent une classe d’exemples éventuellement non cohérents pour l’invariance d’homotopie de la K-théorie algébrique. Comme application, nous montrons que les espaces de Riemann–Zariski relatifs de Temkin satisfont également l’invariance d’homotopie pour la K-théorie sous certaines hypothèses de finitude.

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Accepted:
Published online:
DOI: 10.5802/crmath.786
Classification: 19E08, 19D35
Keywords: K-theory, regularity, homotopy invariance, semi-valuation rings, Riemann–Zariski spaces
Mots-clés : K-théorie, regularité, invariance par homotopie, anneaux de semi-valuation, espaces de Riemann–Zariski

Christian Dahlhausen 1

1 Institut für Mathematik, Fakultät für Mathematik und Informatik, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Christian Dahlhausen. Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 989-1001. doi: 10.5802/crmath.786

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