Grothendieck groups and completions of Gorenstein local rings

TL;DR

The paper addresses when completion preserves rational Grothendieck groups for excellent Gorenstein isolated local rings by linking the isomorphism $G(A)_{\mathbb{Q}} \to G(\widehat{A})_{\mathbb{Q}}$ to a lifting property of maximal Cohen–Macaulay modules and to a Thomason-type classification of $\mathbb{Q}$-subspaces of the stable CM category. It develops a general framework using radical dense subcategories of skeletally small triangulated categories to classify these subspaces and proves equivalences between the isomorphism on Grothendieck groups and existence statements in the stable category; it extends these ideas to non-isolated henselian rings in low dimensions and provides numerous concrete examples. The results connect completion, Grothendieck and Chow groups, and stable CM theory, yielding practical criteria for when completion preserves these invariants and illuminating how these invariants behave under various ring-theoretic settings. Overall, the work offers a robust structural lens for understanding how completion interacts with birational and homological invariants in Gorenstein singularities.

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.

Theorems & Definitions (48)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Definition 1.4
• Proposition 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Corollary 1.10