TITLE: short exact sequences and direct product
QUESTION [9 upvotes]: Let
$$0\longrightarrow L^{(i)}\longrightarrow M^{(i)}\longrightarrow N^{(i)}\longrightarrow 0$$
be a short exact sequence of abelian groups for every index $i$. Clearly if I take finite direct products, then
$$0\longrightarrow \prod_iL^{(i)}\longrightarrow\prod_i M^{(i)}\longrightarrow \prod_iN^{(i)}\longrightarrow 0$$
is a short exact sequence. But what about infinite direct product? Is the exactness preserved?

REPLY [14 votes]: In a general (abelian) category, the product of epimorphisms (if it exists!) may not be an epimorphism – that is why we sometimes assume (AB4*) as an extra axiom. As it turns out, the category of $R$-modules (for any ring $R$) always satisfies the (AB4*) axiom.
Proposition. If the (AB4*) axiom is satisfied in an abelian category, then products of short exact sequences are short exact sequences.
Proof. Consider short exact sequences
$$0 \longrightarrow L^{(i)} \longrightarrow M^{(i)} \longrightarrow N^{(i)} \longrightarrow 0$$
By general abstract nonsense, one can show that the kernel of a product is the product of the kernels, so we have a left exact sequence
$$0 \longrightarrow \prod_i L^{(i)} \longrightarrow \prod_i M^{(i)} \longrightarrow \prod_i N^{(i)}$$
but by the (AB4*) assumption, the last homomorphism is an epimorphism, so we in fact have a short exact sequence.