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

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.

💡 Research Summary

The paper addresses a subtle question in algebraic K‑theory: whether the natural map induced by completion on Grothendieck groups of finitely generated modules, after tensoring with the rational numbers, is always injective. For a Noetherian local ring (A) with maximal ideal (\mathfrak m), the (\mathfrak m)‑adic completion (\widehat A) comes equipped with a canonical homomorphism (A\to\widehat A). This induces a group homomorphism \