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

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.