On injectivity of maps between Grothendieck groups induced by completion

TL;DR

This paper provides a counterexample showing that the natural map between Grothendieck groups induced by completion can fail to be injective, raising questions about the kernel of this map in algebraic geometry.

Contribution

It constructs a specific local normal domain where the Grothendieck group map is not injective, highlighting limitations of completion in algebraic K-theory.

Findings

Counterexample of non-injective Grothendieck group map

Raises questions about the kernel of the map

Implications for algebraic K-theory and completion

Abstract

We give an example of a local normal domain $R$ such that the map of Grothendieck groups $\G(R) \to \G(\hat R)$ is not injective. We also raise some questions about the kernel of that map.