On page 55 of Weibel's Introduction to homological algebra the following passage appears:



> 
> Here are two consequences that use the fact that homology commutes with arbitrary direct sums of chain complexes
> 
> 
> 


I understand why homology commutes with arbitrary direct sums when the direct sum of a collection of monics is a monic (i.e the direct sum functor is exact) but I was under the impression that there were abelian categories where the direct sum functor is not exact. After a bit of thought, I realised that I don't know an example of an abelian category in which the coproduct functor is not exact.


Sheaves of abelian groups on a fixed topological space give an example of an abelian category in which the product functor is not exact.



> 
> Question 1: Is the passage from Weibel's book correct? If so, then why?
> 
> 
> Question 2: Is there an example of an abelian category where the direct sum functor is not exact?
> 
> 
> 



In an arbitrary abelian category, does chain complex homology commute with coproduct?
I couldn't think of a natural example of an abelian category in which direct sums are not exact (I think this is called axiom AB4). For example, sheaves of abelian groups and R-modules both have this property. However there are natural examples of abelian categories where direct products are not exact (i.e. not satisfying AB4\*), for example, the category of abelian sheaves on a space.


Taking the opposite category of such a category will then give an example of a category not satisfying AB4 (albeit, not a very nice one).


Once you have such an example, homology of chain complexes in this category will not commute with direct sum:


if $A\_i \to B\_i$ is a sequence of monos such that $\bigoplus (f\_i :A\_i \to B\_i)$ is not a mono, then consider the sequence of two-term complexes


$A\_i \to B\_i$.


$H^0$ of each of these complexes is zero, but $H^0$ of the direct sum is the kernel of $\bigoplus f\_i$.


Here is one way to see that Sh(X) does not satisfy AB4\* (probably not the easiest). Assume for simplicity X = [0,1]. Take a finite open cover, $\mathcal U\_i$ of X by balls of radius $1/i$. Let $A\_i$ be the sheaf


$\prod \_{U \in \mathcal U\_i} j\_{U!} \mathbb Z\_U$.


This has an epimorphism to $\mathbb Z\_X$, but the direct product of all of them together is not epimorphic: taking sections over any open set $V$ will kill off any $A\_i$ when no $1/i$-ball contains $V$.


I hope this is correct!


