Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and only if $h$ is a monomorphism?

It is clear that for any category of modules this is true.

share|improve this question

2 Answers 2

$f$ epi $\Leftrightarrow$ $\mathrm{im}(f)=B$ $\Leftrightarrow$ $\ker(g)=B$ $\Leftrightarrow$ $g=0$ $\Leftrightarrow$ $\mathrm{im}(g)=0$ $\Leftrightarrow$ $\ker(h)=0$ $\Leftrightarrow$ $h$ mono.

share|improve this answer

Let $f$ be an epimorphism. If the image of $f$ is the kernel of $g$ then $g$ is constant and then the image of $g$ being the kernel of $h$ is trivial which is enough to have $h$ as an monomorphism.

Let $h$ be a monomorphism, then the image of $g$ is constant, hence $f$ is surjective.

share|improve this answer
1  
"the image of $g$ is constant" and "$f$ is surjective" are not precise. –  Martin Brandenburg Feb 21 at 8:53
    
"in the background" or "little under the surface" are the hypothesis @MartinBrandenburg –  janmarqz Feb 21 at 12:11

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.