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For a Noetherian local ring $(R,m)$, and a finite $R$-module $M$ with $\operatorname{depth} M=t,$ type of $M$ is defined to be $r(M):=dim_{R/m}Ext^t \ (R/m, M).$

Is there a characterization of $r(M)$ by local cohomology instead of $Ext$?

If so, can you give a reference?

Thank you.

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I think (it is well-known that) $r(M) = \dim \mathrm{Soc}(H^t_{\mathfrak{m}}(M)) = \ell (\mathrm{Hom}(R/\mathfrak{m}, H^t_{\mathfrak{m}}(M)))$.

Indeed, choose an $M$-regular sequence $x_1, \ldots, x_t$ of $M$. We have $$\mathrm{Ext}^t_R(R/\mathfrak{m}, M) \cong \mathrm{Hom}(R/\mathfrak{m}, M/(x_1, \ldots, x_t)M).$$ We proceed by induction on $t$ that $\mathrm{Hom}(R/\mathfrak{m}, H^t_{\mathfrak{m}}(M)) \cong \mathrm{Hom}(R/\mathfrak{m}, M/(x_1, \ldots, x_t)M)$. The case $t = 0$ follows from $\mathrm{Hom}(R/\mathfrak{m}, H^0_{\mathfrak{m}}(M)) \cong 0:_M \mathfrak{m} \cong \mathrm{Hom}(R/\mathfrak{m}, M)$. For $t>0$, consider the short exact sequence $$0 \to M \overset{x_1}{\to} M \to M/x_1M \to 0.$$ Apply local cohomology functor we have the exact sequence with note that $\mathrm{depth} M = t$ and $\mathrm{depth} M/x_1M = t-1$ $$0 \to H^{t-1}_{\mathfrak{m}}(M/x_1M) \to H^t_{\mathfrak{m}}(M) \overset{x_1}{\to} H^t_{\mathfrak{m}}(M) \to \cdots.$$ Thus $H^{t-1}_{\mathfrak{m}}(M/x_1M) \cong 0:_{H^t_{\mathfrak{m}}(M)}x_1$. Therefore $$\mathrm{Hom}(R/\mathfrak{m}, H^{t-1}_{\mathfrak{m}}(M/x_1M)) \cong (0:_{H^t_{\mathfrak{m}}(M)}x_1) : \mathfrak{m} = 0:_{H^t_{\mathfrak{m}}(M)} \mathfrak{m} \cong \mathrm{Hom}(R/\mathfrak{m}, H^t_{\mathfrak{m}}(M)).$$ Now the claim follows from the inductive hypothesis.

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