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

Say you have some longs exact sequences of modules

0⟶M1ϕ1⟶M2ϕ2⟶M3ϕ3⟶M4ϕ4⟶⋯

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

0⟶N1⟶M1⟶N′1⟶0⟶N2⟶M2⟶N′2⟶0⟶⋯?

If so, how does this work? Merci.

**Answer**

You can think of the long exact sequence

0⟶M1ϕ1⟶M2ϕ2⟶M3ϕ3⟶M4ϕ4⟶⋯

as a collection of short exact sequences

0⟶M1ϕ1⟶M2ϕ2⟶Image(ϕ2)⟶0

0⟶Coker(ϕ2)ϕ3⟶M4ϕ4⟶Image(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*