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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.

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  • $\begingroup$ You can get rid of that $N^{*}$ by using local duality and the fact that since $M$ is finitely generated there's an isomorphism $\text{Tor}_{i}(N,\text{Hom}(H_{\mathfrak{m}}^{d}(N),E(k)))\simeq \text{Hom}(\text{Ext}^{i}(M,H_{\mathfrak{m}}^{d}(N)),E(k))$. $\endgroup$
    – Zeek
    Commented Jul 24, 2020 at 9:41
  • $\begingroup$ @zeek: thanks for catching that ... I updated it in my post $\endgroup$
    – user521337
    Commented Jul 25, 2020 at 6:43

1 Answer 1

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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.

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  • $\begingroup$ 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. $\endgroup$ Commented Jul 25, 2020 at 15:47

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