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I'm working on exercise 1.6.H.a) of Ravi Vakil's algebraic geometry course notes. I'm aware that a question was posted on the same topic before (Prove the FHHF theorem using as much abstract non-sense as possible), but it looks like there weren't any responses. I followed the hints up to showing that there is an epimorphism $F(\text{im} d^i) \rightarrow \text{im}F(d^i)$. Using the next part of the hint, I had one right exact sequence given by applying the functor $F$ to the exact sequence for the complex $C^\cdot$ given in the hint and an exact sequence by directly applying the exact sequence to the complex $FC^\cdot$.

Given the epimorphism $F(\text{im} d^i) \rightarrow \text{im}F(d^i)$ and an isomorphism $F(\text{coker}d^{i - 1}) \rightarrow \text{coker}F(d^{i - 1})$, I got a square of maps. However, I was unable to show that it is commutative (the problem would be solved in that case). Is there any other way to approach this problem? Does the fact that we have an epimorphism $F(\text{im} d^i) \rightarrow \text{im}F(d^i)$ help?

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mahimahi
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  • Are you happy to check this in the category of $A$-modules? Without this it's a little awkward -- my guess is that most people who get through this section in Vakil probably haven't checked that all the abelian category stuff "works" and that's probably good. – Hoot Jul 09 '15 at 03:57
  • My question was specifically with general abelian categories, but I think I'll try doing that if I can't figure out how things work in general abelian categories. Just wondering, why is it useful to consider things in general abelian categories rather than just pretending that we're working with $A$-modules? – mahimahi Jul 09 '15 at 05:42
  • Please consider typing the exercise to your question so that nobody has to download that whole pdf file. Also, is "algebraic geometry" really an appropriate tag, since your question is only about homological algebra? – Martin Brandenburg Jul 09 '15 at 08:58
  • I misread the post I linked to and thought that it had an "algebraic geometry" tag. I removed the tag and put the link at the beginning of the question in place of the link to the pdf. – mahimahi Jul 09 '15 at 10:11

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