When does homology commutes with arbitrary direct sums

Question

Is it necessary to have the criteria that the direct sum of a collection of monics is a monic, to show that homology commutes with arbitrary direct sums? Because when I tried to prove the result, I really didn't use this criteria and I don't see how it can be used ....can someone please help me understand this ?

Answer 1

In an abelian category satisfying AB5, homology preserves all direct sums. Indeed, a functor preserves all direct sums as soon as it preserves finite direct sums and filtered colimits, and a functor is additive if and only if it preserves finite direct sums, so it suffices to verify the following:

1. $\ker$ commutes with filtered colimits.
2. $\operatorname{coker}$ commutes with filtered colimits.

But the first is a special case of the AB5 axiom, and the second is automatic, so homology indeed preserves all direct sums. In particular, this is true in any category of modules (over a ring).

However, you can prove the claim with weaker assumptions than AB5. It suffices to assume that direct sums of exact sequences are exact sequences, which is equivalent to assuming that direct sums of monomorphisms are monomorphisms (because direct sums are automatically right exact). Conversely, if homology commutes with direct sums, then direct sums of exact sequences must be exact sequences, so this is also a necessary condition.

Answer 2

Comments for Zhen Lin's answer (because I have not enough reputation):

Actually the homology functor $H_n\colon Ch(A) \to A$, where $A$ is an abelian category, commutes with arbitrary direct limit, iff $A$ satisfies axiom Ab4), i.e. $A$ is cocomplete and a direct sum of a monic is monic.

You should be careful about Weibel's book. There is an error in page 55 says $H_n$ commutes with arbitrary direct sums when $A$ is an arbitrary abelian category, but he has corrected it in his webpage.