How does one split up a long exact sequence into short exact sequences?
Say you have some longs exact sequences of modules
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
If so, how does this work? Merci.
You can think of the long exact sequence
as a collection of short exact sequences
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.