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

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

2007-07-04

arxiv.org/abs/0707.0547v1

Mathematics

Commutative Algebra

15pages

Scientific paper

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.

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