Re: Name of concept?
Marco Grandis <grandis <at> dima.unige.it>
2005-10-03 15:15:27 GMT
Yes, I like this symmetric presentation.
As to "Notice..." (where you mean ... meet im(d), of course), I
would add the following.
The condition dd = 0 is linked with aspects whose relevance is
Without it, many systems of use in homological algebra would loose
any reasonable notion of "canonical isomorphism", and one should be
extremely prudent in working with induced morphisms and Noether
Let us start (in Ab, or any abelian category) with a sublattice L
of subobjects of a given object (necessarily modular) and consider
the subquotients having numerator and denominator in L. Then, the
canonical isomorphisms among these subquotients (induced by the
identity) are closed under composition *if and only if* L is
- Within this restriction, being "canonically isomorphic
subquotients" has a precise meaning: there is a well-determined
canonical isomorphism linking them.
- Without this restriction, composing canonical isomorphisms can
yield different isomorphisms between two given subquotients. Working
up to canonical isomorphism, as commonly done in homological algebra,
could easily lead to errors.
(For instance, it is easy to construct such a situation for