- #1
gentsagree
- 96
- 1
[tex]0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0[/tex] 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 [itex]C\rightarrow 0[/itex] to be the whole of C?
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 [itex]C\rightarrow 0[/itex] to be the whole of C?