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I'm working though Weibel right now, and I'm getting stuck on Exercise 1.3.6:

If $0 \to A \to B \to C \to 0$ is an exact sequence of double complexes, show that $0 \to Tot(A) \to Tot(B) \to Tot(C) \to 0$ is exact.

Obviously each $0 \to A_{p,q} \to B_{p,q} \to C_{p,q} \to 0$ is exact, and I want to just take the direct sum (product?), but I don't think it preserves exactness in an abelian category without AB4, AB4*.

I think there's gotta be more data in the original exact sequence of complexes, that I'm not making use of, but I can't figure out what it is. Any hints?

Henry Swanson
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    In my copy, it says “double complexes of modules”, so you have AB4 and AB4*. – Jeremy Rickard Nov 19 '17 at 20:17
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    Well, guess I gotta work on my reading comprehension. Thanks! This entire chapter was "yeah, we'll do it in R-mod but it works everywhere because of Freyd (which we haven't proven)", and I guess that threw me off. – Henry Swanson Nov 19 '17 at 20:39

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