# Tor commutes with direct limits

Weibel has shown that the filtered colimit functor $$\varinjlim :(RMod)^I \rightarrow (RMod)$$ where $$I$$ is a filtered category, is exact. In corollary 2.6.16, pg 58 he claims

Corollarry 2.6.16. If $$F:RMod \rightarrow B$$, $$B$$ an abelian category. and $$F$$ is a left adjoint, then $$L_*F(\varinjlim A_i) \simeq \varinjlim L_*F(A)$$.

I am struggling to prove this:

Problem: How does one construct a projective resolution of $$\varinjlim A_i$$ that respects the preojctive resolution of $$P_i \rightarrow A_i$$?

I thought $$\varinjlim P_i$$ would be projective, but this turns out to be false as given in comments.

• ${\mathbb Q}=\lim_{\to}{1\over n}{\mathbb Z}$ is not a projective ${\mathbb Z}$-module. – peter a g May 7 at 17:10
• @peterag then do you know how one deduces 2.6.16? – Bryan Shih May 7 at 17:11
• This MO post might be helpful. – JHF May 8 at 15:39
• Perhaps Weibel is being too fast? See also @PedroTamaroff's answer to "math.stackexchange.com/questions/311615/…": the argument uses that the filtered direct limit of projectives is flat to show that filtered limits and tor commute (Weibel's Cor 2.6.17). – peter a g May 8 at 17:40
• And though not pertinent for this question, the following are (links to) Weibel's lists of errata for the text: sites.math.rutgers.edu/~weibel/Hbook-corrections.html – peter a g May 8 at 18:00