# Proving a duality between Ext and Tor for maximal Cohen-Macaulay modules over Gorenstein ring

Let $$(R,\mathfrak m, k)$$ be a local complete Gorenstein ring of dimension $$d$$. Let $$M,N$$ are maximal Cohen-Macaulay modules (i.e. have depth equal to $$d$$) that are locally free on the punctured spectrum (i.e. $$M_P, N_P$$ are free over $$R_P$$ for every non-maximal prime ideal $$P$$ of $$R$$) . Let $$E(k)$$ be the injective hull of the residue field $$k$$.

Then, how to prove that

$$\text{Ext}^d_R( \text{Tor}_i^R(M,N^*), R)\cong \text{Ext}^{d+i}_R(M,N),\forall i\ge 1$$ ?

Here $$(-)^*:=\text{Hom}(-, R)$$

My thoughts: Let us write $$(-)^{\lor}:=\text{Hom}(-,E(k))$$. Since $$M,N$$ are locally free on the punctured spectrum, so $$\text{Tor}_i^R(M,N^*)$$ has finite length for every $$i>0$$. So $$H^0_{\mathfrak m}(\text{Tor}_i^R(M,N^*))\cong \text{Tor}_i^R(M,N^*)$$ . So by local duality, we get

$$\text{Ext}^d_R( \text{Tor}_i^R(M,N^*), R)\cong (H^0_{\mathfrak m}(\text{Tor}_i^R(M,N^*))^{\lor} \cong ( \text{Tor}_i^R(M,N^*))^{\lor}\cong \text{Tor}_i^R(M, H^d_{\mathfrak m}(N)^{\lor})^{\lor} \cong \text{Ext}^i_R(M, H^d_{\mathfrak m}(N))$$

So basically we're trying to prove $$\text{Ext}^i_R(M, H^d_{\mathfrak m}(N))\cong \text {Ext}^{d+i}_R(M,N),\forall i\ge 1$$.

Also note that for any module $$M$$, we have a stable isomorphism $$syz^2 \text{Tr} M \cong M^*$$ , where $$syz^2(-)$$ denotes second syzygy and $$\text{Tr }(-)$$ denotes Auslander transpose. So, $$\text{Tor}_i^R(M,N^*)\cong \text{Tor}_{i+2}^R(M, \text {Tr }N)$$ .

But I'm unable to simplify things further.

One key point that might be useful is that over Gorenstein local rings, maximal Cohen-Macaulay modules are reflexive and their duals are again maximal Cohen-Macaulay.

• You can get rid of that $N^{*}$ by using local duality and the fact that since $M$ is finitely generated there's an isomorphism $\text{Tor}_{i}(N,\text{Hom}(H_{\mathfrak{m}}^{d}(N),E(k)))\simeq \text{Hom}(\text{Ext}^{i}(M,H_{\mathfrak{m}}^{d}(N)),E(k))$.
In the proof of Theorem 3.2 in this survey of local cohomology by Schenzel it is shown that there is a spectral sequence $$E_{2}^{i,j}=\text{Ext}_{R}^{i}(M,H_{\mathfrak{m}}^{j}(N))\Rightarrow \text{Ext}_{R}^{n}(M,N)$$ (I do not think the assumptions he makes in the theorem are used to prove the existence of this sequence). Since $$N$$ is CM this spectral sequence is concentrated in the $$j=d$$ column so collapses immediately giving isomorphisms $$Ext_{R}^{i}(M,H_{\mathfrak{m}}^{d}(N))\simeq \text{Ext}_{R}^{i+d}(M,N).$$ This gives the last isomorphism in your post.
• More directly, there is also a Grothendieck spectral sequence $\operatorname{Ext}^p_R(\operatorname{Tor}^R_q(M,N^*),R) \Rightarrow \operatorname{Ext}^{p+q}_R(M,N^{**})$ which collapses (taking $q>0$) similarly. Commented Jul 25, 2020 at 15:47