The Stacks project

Lemma 56.4.6. Let $A$ and $B$ be Noetherian rings. Let $F : \text{Mod}^{fg}_ A \to \text{Mod}^{fg}_ B$ be a functor. Then $F$ extends uniquely to a functor $F' : \text{Mod}_ A \to \text{Mod}_ B$ which commutes with filtered colimits. If $F$ is additive, then $F'$ is additive and commutes with arbitrary direct sums. If $F$ is exact, left exact, or right exact, so is $F'$.

Proof. See Lemmas 56.4.3 and 56.4.5. Also, use the finite $A$-modules are finitely presented $A$-modules, see Algebra, Lemma 10.31.4, and use that Noetherian rings are coherent, see Algebra, Lemma 10.90.5. $\square$

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