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Bruns and Herzog in their book Cohen-Macaulay Rings, page 128 consider a local Noetherian ring $(R,m,k)$, an $R$-module $M$ and they define the functor $\Gamma_m(\cdot)$ as $\Gamma_m(M) = \varinjlim \operatorname{Hom}_R(R/m^s,M)$. Proposition 3.5.1 says that this functor is left exact.

Question: Since $\operatorname{Hom}_R(R/m^s,\cdot)$ is a left exact functor and $\varinjlim$ is an exact functor, does it not follow immediately that $\varinjlim \operatorname{Hom}_R(R/m^s,\cdot)$ is a left exact functor? Why do we need a proof as the one given by B&H?

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I think you're right But the proof of B&H is an elementary proof that don't use exactness of $\varinjlim$ and it is kind of instructive.

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