Equivalences of stable categories of Gorenstein local rings

TL;DR

The paper develops a framework to realize triangle equivalences between stable categories of maximal Cohen–Macaulay modules over Gorenstein local rings by exploiting pointed étale neighborhoods and Grothendieck-group invariants. Under assumptions such as isolated singularities, domain completions, and finitely generated Grothendieck groups $G(\widehat{A})$, it shows how CM-stable categories stabilize along directed systems of étale neighborhoods and transfer to henselizations, yielding many new equivalences between $\\underline{CM}(A)$ and $\\underline{CM}(B)$. It provides a broad array of explicit examples (symmetric semigroup rings, invariant rings, simple singularities, quadratic forms) where these equivalences occur, and a technical result ensuring that henselizations of such rings are not of finite type, which underpins the construction. The work expands the catalog of known CM-stable equivalences and offers tools for generating further instances in this area of singularity theory.

Abstract

In this paper we give bountiful examples of Gorenstein local rings $A$ and $B$ such that there is a triangle equivalence between the stable categories \underline{CM}($A$), \underline{CM}($B$).

Theorems & Definitions (19)

• Remark 1.1
• Remark 1.2
• Theorem 2.4
• proof
• Theorem 2.5
• Lemma 2.6
• proof
• proof : Proof of Theorem \ref{['main-onto']}
• Corollary 2.7
• proof