Grothendieck groups and completions of Gorenstein local rings

Abstract

Let $(A,\mathfrak{m})$ be an excellent Gorenstein local ring of dimension $d \geq 2$ which is an isolated singularity. Let $\widehat{A}$ denote the completion of $A$. If $G(A)$ is the Grothendieck group of $A$ then by $G(A)_\mathbb{Q}$ we denote $G(A)\otimes_\mathbb{Z} \mathbb{Q}$. We prove that the natural map $G(A)_\mathbb{Q} \rightarrow G(\widehat{A})_\mathbb{Q}$ is an isomorphism if and only if for any maximal Cohen-Macaulay (= MCM) $\widehat{A}$-module $M$ there exists an MCM $A$-module $N$ and integers $r \geq 1$ and $s \geq 0$ (depending on $M$) such that $M^r\oplus \widehat{A^s} \cong \widehat{N}$. An essential ingredient is the classification of $\mathbb{Q}$-subspaces of $G(\mathcal{C})_\mathbb{Q}$ (here $\mathcal{C}$ is a skelletaly small triangulated category) in terms of certain dense subcategories of $\mathcal{C}$. We also give criterion for a Henselian Gorenstein ring $B$ (not an isolated singularity) such that the natural map $G(B)_\mathbb{Q} \rightarrow G(\widehat{B})_\mathbb{Q}$ is an isomorphism ( when $\dim B = 2, 3$). We give many examples where our result holds.