On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings

TL;DR

This paper investigates how the stable category of maximal Cohen-Macaulay modules over Gorenstein rings determines key properties of the rings, such as being a complete intersection, dimension, and singularity type, under triangulated equivalences.

Contribution

It establishes that triangulated equivalences of stable categories preserve important ring invariants like codimension, dimension, and singularity properties for Gorenstein local rings.

Findings

Complete intersection property is preserved under stable category equivalence.

Ring dimension is preserved for Henselian non-hypersurface Gorenstein rings.

Isolated singularity property is preserved under stable category equivalence.

Abstract

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $CMS(A)$ be its stable category of maximal CM $A$-modules. Suppose $CMS(A) \cong CMS(B)$ as triangulated categories. Then we show (1) If $A$ is a complete intersection of codimension $c$ then so is $B$. (2) If $A, B$ are Henselian and not hypersurfaces then $\dim A = \dim B$. (3) If $A, B$ are Henselian and $A$ is an isolated singularity then so is $B$. We also give some applications of our results. It should be remarked that if $R,S$ are complete CM but not necessarily Gorenstein and if there is an triangle isomorphism between the singularity categories of $R$ and $S$ then it is possible that $\dim R - \dim S$ is odd, see M.~Kalck; Adv. Math. 390 (2021), Paper No. 107913.