Let $A\stackrel{f}{\to}B \stackrel{g}{\to}C$ be a complex in an abelian category, I.e. $gf=0$.
It is easy to see that there exists morphism $im(f) \to ker(g).$
Could you have a proof for it being a monomorphism?
Let $A\stackrel{f}{\to}B \stackrel{g}{\to}C$ be a complex in an abelian category, I.e. $gf=0$.
It is easy to see that there exists morphism $im(f) \to ker(g).$
Could you have a proof for it being a monomorphism?
It follows from the fact that
$ im(f) \to ker(g) \to B$
is the monomorphism
$ im(f) \to B $
In other words, $im(f) \to ker(g)$ isn't just a morphism of objects, but it is a morphism of subobjects of $B$.
This is, of course, rather trivial if you allow yourself to prove theorems in abelian categories by diagram chase, since $im(f) \subseteq ker(g) \subseteq B$.