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Suppose $R$ is a Noetherian local ring, and $\kappa$ its residue field. For $R$ module $M$, we can consider the module $$N:=\kappa \otimes_S RHom(\kappa,M)$$ where $S$ is the derived ring of endomorphisms of $\kappa$, namely $RHom(\kappa,\kappa)$. There is a natural map from $N$ to $M$, under what conditions is $N$ the local cohomology of $M$? I'm fairly sure it's true when $R$ is regular (I think i have a proof) but i suspect it holds much more generally. Note the tensor product is of course also derived.

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