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Here is the question from Rotman, verbatim:

A sequence S'_*\stackrel{f}{\to} S_* \stackrel{g}{\to} S''_* is exact in Comp if and only if S'_{n}*\stackrel{f_n}{\to} S_n \stackrel{g_n}{\to} S''_{n} is exact in Ab for every $n\in \mathbb{Z}$

(Note here Comp and Ab are the categories of chain complexes and abelian groups respectively.

I am slightly confused - I thought this is the definition of an exact sequence of chain complexes. Indeed this is what Massey appears to say (Definition 2.6)

What am I missing here?

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    Just an observation: The category of complexes is an abelian category, so there is an intrinsic notion of exactness and it turns out to be what you'd expect it to be. As @Rasmus pointed out, in order to check that the category of complexes is abelian, you have to think about what kernel, cokernel of a chain map are - they are the obvious candidates just check the factorization properties. This also follows from the fact that $\operatorname{Comp}$ can be written as a functor category (with some care), so kernels and cokernels (and all limits and colimits for that matter) can be taken pointwise.2011-03-01

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