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 $r(M):=dim\_{R/m}Ext^t \ (R/m, M).$ 



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> Is there a characterization of $r(M)$ by *local cohomology* instead of $Ext$?
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If so, can you give a reference? 


Thank you. 



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


