On Injectivity of Maps between Grothendieck Groups Induced by Completion

摘要

Let (R,m,k) be a local ring and let R be the m-adic completion of R. Let M(R) be the category of finitely generated R-modules. The Grothendieck group of finitely generated modules over R is defined as G(R) = (⊕_(M∈M(R)) Z[M])/(<[M_2] - [M_1] - [M_3]|0 → M_1 → M_2 → M_3 → 0 is exact>_. Kamoi and Kurano [KKu] studied injectivity of the map G(R) → G(R) induced by flat base-change.