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.
Please help.
Best Answer
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.