Question on exact sequence

[Michael Barr <barr@math.mcgill.ca>]

I have recently discovered a curious fact about abelian categories.
First, let me briefly describe the well-known snake lemma.  If we have a
commutative diagram with exact rows (there are variations without the 0
at the left end of the top and without the 0 at the right end of the
bottom, but here is the strongest form)

      0 ---> A ----> B ----> C ----> 0
             |       |       |
             |       |       |
             |f      |g      |h
             |       |       |
             v       v       v
      0 ---> A' ---> B' ---> C' ---> 0

then there is an exact sequence
  0 --> ker f --> ker g --> ker h --> cok f --> cok g --> cok h --> 0

The curious discovery is that you have any pair of composable maps f: A
--> B and h: B --> C and you form the diagram (with g = hf)
                 1       f
             A ----> A ----> B
             |       |       |
             |       |       |
             |f      |g      |h
             |       |       |
             v       v       v
             B ----> C ----> C
                 h       1
you get the same exact sequence.  So I would imagine that there must be
a "master theorem" of which these are two cases.  Does anyone know what
it says?  The connecting map here is just the inclusion of ker h into B
followed by the projection on cok f.

Michael



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