Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $\mathcal{A}$ be an $\mathbf{AB5}$ category with enough projectives and let $F:\mathcal{A}\rightarrow\mathcal{B}$ be a right exact functor into abelian category that commutes with filtered colimits. What reasonable assumptions should one imposed on $\mathcal{A}$ and $F$ to obtain that letf derived functors of $F$ commute with filtered colimits?

share|improve this question

2 Answers 2

When $\mathcal{B}$ is also AB5, this is the case iff filtered colimits of projectives are $F$-acyclic:

"$\Rightarrow$" If $\{X_i\}$ is a filtered system of projectives, then $$(L^* F)(\mathrm{colim}_i X_i) = \mathrm{colim}_i (L^* F)(X_i) = 0,$$ i.e. $\mathrm{colim}_i X_i$ is $F$-acyclic.

"$\Leftarrow$" If $\{X_i\}$ is some filtered system, choose for each $X_i$ some projective resolution $(P_{\bullet})_i \to X_i$. By assumption and since $\mathcal{A}$ is AB5, $\mathrm{colim} (P_{\bullet})_i$ is a $F$-acylic resolution of $\mathrm{colim}_i X_i$, so that we can use it to compute the Left derived functors. It follows (since $\mathcal{B}$ is AB5)

$(L^* F)(\mathrm{colim}_i X_i) = H_*(\mathrm{colim}_i (P_{\bullet})_i) = \mathrm{colim}_i H_*(P_{\bullet})_i) = \mathrm{colim}_i (L^* F)(X_i)$

and this is induced by the canonical morphism $\leftarrow$. $\square$

For example, the tensor product $M \otimes_R - : \mathrm{Mod}(R) \to \mathrm{Ab}$ satisfies this assumption, since filtered colimits of projectives are flat and therefore acyclic with respect to the tensor product. It follows that $\mathrm{Tor}_*(M,-)$ commutes with filtered colimits.

share|improve this answer
    
Martin thank you for your answer. Do you know any usefull criteria when acyclic objects are closed under taking filtered colimits? –  Anonymous May 22 '12 at 17:09
    
I don't, but I think without any specific choice of $F$ there is no general criterion. –  Martin Brandenburg May 22 '12 at 17:47
    
I wish know a reference about how the morphisms of the filtred system have a (coherent) lifting to a filtred system of projective resolutions. –  Buschi Sergio May 22 '12 at 20:18
    
@Buschi: What is unclear? –  Martin Brandenburg May 23 '12 at 6:18
    
@Brandenburg You deliver the colimit $C_n:=colimi_{i\in I} P_{n, i}$, I dont understand (my ignorance probabily) how are defined the morphism in the $I$-diagram $(P_{n,i})_{i\in I}$ , your proof builds only the objects of this diagram. –  Buschi Sergio May 23 '12 at 11:58

This is only a comment on some aspect of the Brandenburg answere. I wish elucidate the aspect about the $I$-naturality of projective resolutions:

considerind a example of a single morphism $f: A\to B$ and two projective resolution $P_\ast\to A$ and $Q_\ast\to B$ then is a well knowed fundamental lemma of homological algebrathat exist (homotopically unique) a extentions of $f$ to the (augmented) chain complexes. In the basic homological algebra text there is also the example of a resolution of exat sequences.

But what if insted a single morphism (or a exat sequence) we consider a small diagram?

From Weibel, "An introduction to homological algebra", 2.3.13 on p.43 (I see it from link text) follow that $\mathcal{A}^I$ has enought projectives, but isnt clear if for a projective $P\in \mathcal{A}^I$ each $P(i)\in \mathcal{A}$ is projective as we need in the context of the above Brandenburg proof

(this is true if for any $i\in I$ the right Kan extention of the $i$-valutation $v(i): \mathcal{A}^I\to \mathcal{A}$ is exact, I dont know if this follow from the "$I$ is filtrant" hypothesis).

From the book "Theory of Categories (BArry Mitchell) cor.7.6 p. 138, let $T_i: \mathcal{A}^I\to \mathcal{A}$ ($i\in I$) the $i$-valuation, and $S_i$ its left adjuction (the left Kan extention), now for a projective $P\in \mathcal{A}$ the object $S_i(P)(j),\ j\in I$ is a sum of copies of $P$ (see the Weibel reference above) then is projective. THe above corollary assert that projectives of $\mathcal{A}^I$ are objects of the form $\bigoplus_{i\in I}S_i(P_i)$ (where $P_i\in \mathcal{A}$ is a prjective) and all its retracts.

This is enought for ensure the existence a projective resolution of a $(X_i)_i\in \mathcal{A}^I$ with projective arguments.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.