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Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field. Let $X$ be a complex of $R$-modules with finitely generated ...

Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?