2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132938In 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.15pagesCommutative AlgebraK-Theory and Homology13D15, 19A49text