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

Let $(A,\mathfrak{m}),(B,\mathfrak{n})$ be Gorenstein local rings and let $\underset{‾}{CM}(A)$ be its stable category of finitely generated maximal Cohen-Macaulay $A$-modules. Suppose we have an equivalence $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:\underset{‾}{CM}(A)\to \underset{‾}{CM}(A)$ as triangulated categories. We show (1) If $A,B$ are not hypersurfaces then $dimA=dimB$. (2) If $M$ is a maximal \CM \ $A$-module then $${\text{curv}}_A(M)={\text{curv}}_B(\mathrm{\Phi }(M)),$$ here ${\text{curv}}_A(M)={\mathrm{lim\hspace{0.17em}sup}}_n\sqrt[n]{\mathrm{\ell }({\text{Tor}}_n^A(M,k))}$. (3) $A$ satisfies Serre’s condition $R_i$ if and only if $B$ satisfies $R_i$. (4) $A$ is a complete intersection on the punctured spectrum of $A$ if and only if $B$ is a complete intersection on the punctured spectrum of $B$. We also show that $\mathrm{\Phi }$ imposes constraints of residue field of $B$ in terms of residue field of $A$ and vice-versa. Finally if $I$ is an ideal in $A$ such that the extended Rees algebra $\mathcal{R}(I)=A[It,t^{-1}]$ is Gorenstein then we construct a triangulated functor $\mathrm{\Psi }\text{\hspace{0.17em}⁣}:{\underset{‾}{CM}}^{\mathbb{Z}}(\mathcal{R}(I))\to \underset{‾}{CM}(A)$ where ${\underset{‾}{CM}}^{\mathbb{Z}}(\mathcal{R}(I))$ is the stable category of all graded maximal Cohen-Macaulay $\mathcal{R}(I)$-modules. We show that $\mathrm{\Psi }$ induces an equivalence ${\underset{‾}{CM}}^{\mathbb{Z}}(\mathcal{R}(I))\mathrm{/}\ker \mathrm{\Psi }\to \underset{‾}{CM}(A)$. We give some applications of this map.