A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective

TL;DR

This paper constructs a specific two-dimensional local ring over complex numbers where the rational Grothendieck group map to its completion is not injective, revealing new insights into algebraic K-theory and completion behavior.

Contribution

It provides the first explicit example of a local ring with a non-injective map between Grothendieck groups after completion, highlighting subtle properties of algebraic K-theory.

Findings

The kernel of the Grothendieck group map is non-zero in the constructed example.

The local ring is two-dimensional, essentially of finite type over C, but not normal.

The example demonstrates that completion can alter algebraic K-theory invariants.

Abstract

In this paper, we construct a local ring $A$ such that the kernel of the map $G_0(A)\subq \to G_0(\hat{A})\subq$ is not zero, where $\hat{A}$ is the comletion of $A$ with respect to the maximal ideal, and $G_0()\subq$ is the Grothendieck group of finitely generated modules with rational coefficient. In our example, $A$ is a two-dimensional local ring which is essentially of finite type over ${\Bbb C}$, but it is not normal.