How do you split a long exact sequence into short exact sequences?

How does one split up a long exact sequence into short exact sequences?

Say you have some longs exact sequences of modules
0M1ϕ1M2ϕ2M3ϕ3M4ϕ4
I’ve read it’s possible to split this into short exact sequences. What exactly does that mean? Would it be written as short exact sequences, one appended to another like
0N1M1N10N2M2N20?
If so, how does this work? Merci.

Answer

You can think of the long exact sequence
0M1ϕ1M2ϕ2M3ϕ3M4ϕ4
as a collection of short exact sequences
0M1ϕ1M2ϕ2Image(ϕ2)0
0Coker(ϕ2)ϕ3M4ϕ4Image(ϕ4)0

where each sequence after the first begins with the relevant cokernel (well, so does the first, but this is just M1) and ends with the relevant image. I have abused notation here by writing ϕn for the maps from the cokernel which where originally from the corresponding module; this is not a serious issue because exactness of the original sequence ensures that the natural maps (defined by sending an equivalence class to the image of a representative) will be well-defined. One could write this as a single long chain like you proposed, but I prefer not to.

Attribution
Source : Link , Question Author : GGGG , Answer Author : Alex Becker

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