Talk:Derived functor

Motivation

The motivation section should be changed. The thing really comes from applying tensor and hom in topology. I'll update it sometime.

The term "Canonical" and more so the term "natural" are used loosely outside of category theory, but since the context here is category theory, shouldn't these be unquoted and made into links, as would be standard for any technical terms? MotherFunctor 04:42, 14 May 2006 (UTC)

0th derived functors

I'd like to add a comment that R⁰F(A)=F(A) and L₀F(A)=F(A), because it's not obvious from the definitions, so it's confusing to beginners. At least, it confused me. I'm new to editting wiki, though, so I'm posting here first to give others a chance to flame me in a less public arena. Adam1729 03:51, 26 November 2006 (UTC)

Actually this is true only if you assume that your functor is left (or right)-exact. But you can construct derived functors without any such assumption, and this is actually quite important to do sometimes (for example, people study the left derived functor of the adic-completion functor despite the fact that it is not exact from either side). 46.117.193.40 (talk) 08:21, 12 November 2010 (UTC)

Additivity

I am not too proficient in homological algebra, so I might be mistaken here. But shouldn't we require the functor F, which we take the derived functors of, to be additive? Because two distinct projective or injective resolutions need not be isomorphic, but only homotopically equivalent, and I don't see why they retain this homotopic equivalence under the functor F unless we know that F is additive. But maybe I'm blind and additivity follows from left/right exactness? Darij (talk) 13:20, 18 February 2011 (UTC)

If F is not additive, then it needs not even take chain complexes to chain complexes (take F:Ab -> Ab that sends each abelian group to Z and each homomorphism to the identity of Z), and it doesn't make sense to take the cohomology of the resulting complex if the composite of differentials aren't 0. I don't think additivity follows from left or right exactness but I can't think of an example. Money is tight (talk) 20:39, 15 June 2011 (UTC)

Functoriality of choice of injective resolutions?

In the section on constructing the (right) derived functors, there is the following aside:

(Technically, to produce well-defined derivatives of F, we would have to fix an injective resolution for every object of A. This choice of injective resolutions then yields functors $R^i F$. Different choices of resolutions yield naturally isomorphic functors, so in the end the choice doesn't really matter.)

Strictly speaking, do we not need to fix a functorial choice of injective resolution for every object of $A$? That is, resolutions $I(X)$ for objects $X$ such that morphisms $X\to Y$ in $\cA$ give rise to morphisms of complexes $I(X)\to I(Y)$ functorially? It's not clear to me how one could show $R^i F$ to be functorial otherwise. (The Godement resolution does give such a functorial choice of resolutions in the event that $\cA$ is the category of sheaves on a variety.)

142.1.158.234 (talk) 19:12, 23 November 2011 (UTC)