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Short exact sequences

  1. Dec 12, 2013 #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?
  2. jcsd
  3. Dec 12, 2013 #2


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    Because the image of [itex]C[/itex] is [itex]\{0\}[/itex], so by definition everything in [itex]C[/itex] is in the kernel.
  4. Dec 12, 2013 #3
    Of course, thank you.
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