K-theory of admissible Zariski-Riemann spaces

TL;DR

This paper investigates the algebraic K-theory of admissible Zariski-Riemann spaces, establishing its equivalence to G-theory and demonstrating homotopy invariance, with an example related to negative K-theory in non-noetherian categories.

Contribution

It introduces a new understanding of the K-theory of Zariski-Riemann spaces, showing its equivalence to G-theory and homotopy invariance, and provides a novel example involving negative K-theory.

Findings

K-theory of admissible Zariski-Riemann spaces is equivalent to G-theory.

K-theory satisfies homotopy invariance.

Negative K-theory can vanish in non-noetherian abelian categories.

Abstract

We study relative algebraic K-theory of admissible Zariski-Riemann spaces and prove that it is equivalent to G-theory and satisfies homotopy invariance. Moreover, we provide an example of a non-noetherian abelian category whose negative K-theory vanishes.