Short Exact Sequences 0→A→B→C→0: Explained

[gentsagree]

$$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is a short exact sequence if the image of any morphism is the kernel of the next morphism.

Thus, the fact that we have the 0 elements at the two ends is said to imply the following:

1. The morphism between A and B is a monomorphism because it has kernel equal the zero-set {0}, since the image of the map from 0 to A is {0}.

2. The morphism between B and C is an epimorphism because its image is the whole of C.

I understand the first point, but not the second. Why do we require the kernel of $C\rightarrow 0$ to be the whole of C?

 

[pasmith]

$$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is a short exact sequence if the image of any morphism is the kernel of the next morphism.

Thus, the fact that we have the 0 elements at the two ends is said to imply the following:

1. The morphism between A and B is a monomorphism because it has kernel equal the zero-set {0}, since the image of the map from 0 to A is {0}.

2. The morphism between B and C is an epimorphism because its image is the whole of C.

I understand the first point, but not the second. Why do we require the kernel of $C\rightarrow 0$ to be the whole of C?

Because the image of $C$ is $\{0\}$, so by definition everything in $C$ is in the kernel.

 

[gentsagree]

Of course, thank you.