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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 thecomletion of $A$ with respect to the maximal ideal, and $G_0()\subq$ is theGrothendieck group of finitely generated modules with rational coefficient. Inour example, $A$ is a two-dimensional local ring which is essentially of finitetype over ${\Bbb C}$, but it is not normal.

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