  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.
