Goursat's lemma

Goursat's lemma

[Michael Barr]

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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Re: Goursat's lemma

[George Janelidze]

Let us define a short exact sequence 0 --> U --> V --> W --> 0 by saying
that U --> V is the kernel of V --> W ("left exactness") and V --> W is the
cokernel of U --> V ("right exactness"). If so, your formulation becomes is
trivial in full generality:

The kernel of a morphism X --> Y (in any pointed category with zero object)
is the pullback of X --> Y and 0 --> Y. Therefore:

(a) D' is the pullback of G --> C and 0 --> C;

(b) the kernel of D --> G --> C is the pullback of D --> G --> C and 0 -->
C;

(c) as follows from (a) and (b), the kernel of D --> G --> C is the pullback
of D --> G and D' --> C;

(d) similarly, the kernel of D' --> G --> C' is the same pullback.

Since this is true in every category with zero object (you don't need any
normality argument), the dual statement is also always true (and everything
with pushouts and cokernels instead of pullbacks and kernels is equally
clear). Moreover (obviously), for the coincidence of the kernels we need
only the "left exactness", and dually, for the coincidence of the cokernels
we need only the "right exactness" of our original sequences.

However, what Jim's says is not true in every pointed Barr exact category
with cokernels: it will be true if and only if every regular epimorphism is
normal (that is, is the cokernel of something, or, equivalently, the
cokernel of its kernel).

George

--------------------------------------------------
From: "Michael Barr" <barr@math.mcgill.ca>
Sent: Friday, August 26, 2011 1:03 AM
To: "Categories list" <categories@mta.ca>
Subject: categories: Goursat's lemma

> 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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