Re: Question on exact sequence

[Marco Grandis <grandis@dima.unige.it>]

Dear Clemens,

Thank you for your comments.

 > Are there generalizations to n composable arrows ?
The lemma I was proposing works for any sequence of composable arrows.

It can be rewritten in a notation similar to yours, but based on
three consecutive arrows  f, g, h.
One would use, alternatively, two kind of 'generalised homologies'

    H'(f,g,h) = Ker(hg) / Im(f),

    H"(f,g,h) = Ker(h) / Im(gf),

where, again,  H/K  is meant as in my previous msg.

 > What about generalizations to non-abelian categories ?

The proof I was mentioning works for Puppe-exact categories, and is
obvious (AFTER one has
constructed the universal model of a sequence of consecutive arrows).
This is the natural setting of distributive homological algebra, that
cannot be developed under the assumption of products.
It would be too long to explain this here; please see my three papers
on this subject, in Cahiers 1984-85
(if interested, of course.)

Best wishes

Marco


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