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This is Problem 3.42 from Rotman's Homological Algebra text.

Claim

Let $ 0 \to B' \to B \to B'' \to 0$ be an exact sequence of left $R$-modules. If the sequences remains exact after tensoring with all finitely presented right $R$-modules then the sequence is pure exact.

Question

I see that this post provides a proof. Even though the poster tried to prove this without direct limits, this proof was never explicitly given.

Can someone please provide a hint for a direct proof without using direct limits, or at least fill in more details that weren't included in this other post?

IsaacR24
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  • Is the exact sequence correct like this? Currently, you have an isomorphism between $B'$ and $B''$. – Tzimmo Nov 05 '23 at 18:46
  • @Tzimmo Sorry for the typo -- just fixed. Same as it appears in the other post. – IsaacR24 Nov 05 '23 at 19:57
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    Should “remains pure exact” be “remains exact”? – Jeremy Rickard Nov 05 '23 at 20:51
  • Yes, sorry again! – IsaacR24 Nov 05 '23 at 22:02
  • What is the motivation for finding a proof that avoids the most natural and easy approach here? – Martin Brandenburg Nov 05 '23 at 22:42
  • Rotman doesn't address direct limits until a later chapter, so I'm curious how to write the proof with only the tools of the first 3 chapters. – IsaacR24 Nov 06 '23 at 00:48
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    In a sense, "direct limits" is just (excellent) language to describe something that can be broken down into more concrete terms. You can unwind a (co)limit based approach and you should be able to obtain a clear proof. – FShrike Nov 06 '23 at 01:27
  • Ok -- I've reviewed the proof in the other post more closely, and see that Chapter 5 addresses this. The proof is quite simple once I understand direct limits. – IsaacR24 Nov 06 '23 at 17:24

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