I learned [here](https://math.stackexchange.com/questions/234753/finitely-generated-modules-in-exact-sequence) that if the outer groups in a short exact sequence are finitely-generated, then the middle group is, too.

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> **Question.** Is there a generalization of this to long-exact sequences?
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Julian Kuelshammer mentioned the [horseshoe lemma](https://en.wikipedia.org/wiki/Horseshoe_lemma) but I'm not really sure how to apply it.