Re: Question on exact sequence

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

That is really fascinating.  I assume that when you wrote A --> B --> B'
--> C, that last should have been C'.  Here is how it arose.  Call a map A
--> B cotorsion if its cokernel is torsion (for a generalized notion of
torsion).  Assuming that torsion objects are closed under extension (and
that a quotient of torsion is torsion) this allows a proof that a
composite of cotorsion maps is cotorsion.

Anyway, thanks.

Michael

On Tue, 10 Nov 2009, Marco Grandis wrote:

> Dear Michael,
>
> The following lemma extends both results.
>
> We have a sequence of consecutive morphisms indexed on the integers
>
> 	...  ---->   An  ---->  An+1  ---->  An+2  ---->  ...
>
> (if your sequence is finite, you extend by zero's). Call  f(n,m)  the
> composite
> from  An  to  Am  (n < m).
>
> Then, writing  H/K  a subquotient  H/(H intersection K),
> there is an unbounded exact sequence of induced morphisms:
>
>   ...  ---->   Ker f(n,n+2) / Im f(n-1,n)
>  	 ---->   Ker f(n+1,n+2) / Im f(n-1,n+1)
>  	 ---->   Ker f(n+1,n+3) / Im f(n,n+1) ---->  ...
>
> where morphisms are alternatively induced by an 'elementary' morphism
> (say An  -->  An+1) or by an identity.
>
> At each step, one increases of one unit the first index in the numerator
> and the second index in the denominator, or the opposite (alternatively);
> after
> two steps, all indices are increased of one unit, and we go along in the same
> way.
>
> -  Your lemma comes out of a sequence  A ----> B ----> C  (extended with
> zeros).
>
> -  Snake's lemma, with your letters, comes out of a sequence of three
> morphisms
> whose total composite is 0
>
>    					 A ----> B ----> B' ----> C
> taking into account that  A' = Ker(B' ----> C')  and  C = Cok(A ----> B).
>
> I like your lemma (and the Snake's). The form above does not look really
> nice.
> Perhaps someone else will find a nicer solution?
>
> However, if one looks at the universal model of a sequence of consecutive
> morphisms,
> in my third paper on Distributive Homological Algebra, Cahiers 26, 1985,
> p.186,
> the exact sequence above is obvious. (Much in the same way as for the
> sequence of
> the Snake Lemma, p. 188, diagrams (10) and (11).) This is how I found it.
>
> Best wishes
>
> Marco
>
>


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