关于Gorenstein环上极大Cohen-Macaulay模的稳定范畴（II）
On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II
- IIT Bombay（印度理工学院孟买分校）
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AI总结：
本文研究Gorenstein局部环上极大Cohen-Macaulay模稳定范畴的三角等价，证明其保持环的维数、曲率、Serre条件和穿孔谱上的完全交性质，并构造扩展Rees代数稳定范畴到原稳定范畴的三角函子及其诱导等价。
AI中文摘要：
设$(A,\mathfrak{m})$和$(B,\mathfrak{n})$为Gorenstein局部环，令$\underline{CM}(A)$为有限生成极大Cohen-Macaulay $A$-模的稳定范畴。假设存在三角范畴等价$\Phi \colon\underline{CM}(A) \rightarrow \underline{CM}(A)$。我们证明：(1) 若$A, B$均非超曲面，则$\dim A = \dim B$。(2) 若$M$是极大Cohen-Macaulay $A$-模，则$$\text{curv}_A(M) = \text{curv}_B (\Phi(M)),$$其中$\text{curv}_A(M) = \limsup_n \sqrt[n]{\ell(\text{Tor}^A_n(M,k))}$。(3) $A$满足Serre条件$R_i$当且仅当$B$满足$R_i$。(4) $A$在$A$的穿孔谱上是完全交当且仅当$B$在$B$的穿孔谱上是完全交。我们还证明$\Phi$对$B$的剩余域施加了以$A$的剩余域为条件的约束，反之亦然。最后，若$I$是$A$中的理想，使得扩展Rees代数$\mathcal{R}(I) = A[It, t^{-1}]$是Gorenstein的，则我们构造一个三角函子$\Psi \colon \underline{CM}^\mathbb{Z}(\mathcal{R}(I)) \rightarrow \underline{CM}(A)$，其中$\underline{CM}^\mathbb{Z}(\mathcal{R}(I))$是所有分次极大Cohen-Macaulay $\mathcal{R}(I)$-模的稳定范畴。我们证明$\Psi$诱导等价$\underline{CM}^\mathbb{Z}(\mathcal{R}(I))/\ker \Psi \rightarrow \underline{CM}(A)$。我们给出该映射的一些应用。
英文摘要：
Let $(A,\mathfrak{m}), (B,\mathfrak{n}) $ be Gorenstein local rings and let $\underline{CM}(A)$ be its stable category of finitely generated maximal Cohen-Macaulay $A$-modules. Suppose we have an equivalence $Φ\colon\underline{CM}(A) \rightarrow \underline{CM}(A)$ as triangulated categories. We show (1) If $A, B$ are not hypersurfaces then $dim A = dim B$. (2) If $M$ is a maximal \CM \ $A$-module then $$\text{curv}_A(M) = \text{curv}_B (Φ(M)),$$ here $\text{curv}_A(M) = \limsup_n \sqrt[n]{\ell(\text{Tor}^A_n(M,k))}$. (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 $Φ$ 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 $Ψ\colon \underline{CM}^\mathbb{Z}(\mathcal{R}(I)) \rightarrow \underline{CM}(A)$ where $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))$ is the stable category of all graded maximal Cohen-Macaulay $\mathcal{R}(I)$-modules. We show that $Ψ$ induces an equivalence $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))/\ker Ψ\rightarrow \underline{CM}(A)$. We give some applications of this map.
