Talk:Resolution (algebra)
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Left and right
[edit]I realise that only sequences going off to the left are defined so far: can anyone suggest a good terminology for both? Hilton and Stammbach describe them as positive acyclic chain complex and positive acyclic cochain complex which seems clumsy. Richard Pinch (talk) 21:49, 13 October 2008 (UTC)
- I've addressed this issue: left/right resolution is most common, resolution/coresolution is also used (as in Jacobson's Basic algebra). Hilton & Stammbach don't seem to actually address resolutions that are neither projective nor injective... RobHar (talk) 23:10, 27 April 2010 (UTC)
- Just to add to the discussion, it seems that in practice the term `projective resolution' almost exclusively refers to resolutions going to the right (see for example the page on the Horseshoe Lemma, which is what brought me here). However, according to this article, the word resolution should only be used for complexes going of the the left, so someone reading this article would conclude that `projective resolutions' by assumption go to the left. Do you think it would be worth the trouble to rewrite the article such that the term `resolution' refers to both left and right resolutions? 93.51.172.34 (talk) 19:52, 17 January 2016 (UTC)
- It seems that you do a confusion between the direction of the arrows, (which is almost always the right for both resolutions and coresolutions) and the direction in which the resolution is constructed, starting from the "resolved" module. I have rewritten the lead for clarifying this, and also the fact that an injective resolution is, in fact, a coresolution. D.Lazard (talk) 10:48, 18 January 2016 (UTC)
- Just to add to the discussion, it seems that in practice the term `projective resolution' almost exclusively refers to resolutions going to the right (see for example the page on the Horseshoe Lemma, which is what brought me here). However, according to this article, the word resolution should only be used for complexes going of the the left, so someone reading this article would conclude that `projective resolutions' by assumption go to the left. Do you think it would be worth the trouble to rewrite the article such that the term `resolution' refers to both left and right resolutions? 93.51.172.34 (talk) 19:52, 17 January 2016 (UTC)
"Projective resolution of a module M is unique up to a chain homotopy, i.e., given two projective resolution P0 → M and P1 → M of M there exists a chain homotopy between them."
Is this right? To my understanding, homotopies relate morphisms of the chains represented by the resolutions, not the resolutions themselves. — Preceding unsigned comment added by 128.100.109.23 (talk) 04:51, 27 February 2014 (UTC)