Goursat's lemma

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

This is something that Lambek used for diagram chasing in his book on
Rings and Modules.  I have discovered a very simple proof of a slight
generalization.

The lemma states (as least Jim stated it this way) that if G is a
submodule of A x B and C = image of G --> A, C' the image of G --> B, D
the kernel of G --> B and D' the kernel of G --> A, then C/D is isomorphic
to C'/D'.

Forget that G is a subgroup of A x B and just suppose you have two exact
sequences 0 --> D' --> G --> C --> 0 and 0 --> D --> G --> C' --> 0, then
the composites D --> G --> C and D' --> G --> C' have the same cokernel.
The dual claim is that they have the same kernel.  But thinking of D and
D' are submodules of G, then the kernels are quite obviously D \cap D' and
the original conclusion follows by duality.

However, the same thing is true for groups (both D and D', being kernels,
are normal in G and hence in C, C').  I see no such simple duality
argument there.

Michael


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