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.

  • 1
    $\begingroup$ ${\mathbb Q}=\lim_{\to}{1\over n}{\mathbb Z}$ is not a projective ${\mathbb Z}$-module. $\endgroup$ – peter a g May 7 at 17:10
  • $\begingroup$ @peterag then do you know how one deduces 2.6.16? $\endgroup$ – Bryan Shih May 7 at 17:11
  • $\begingroup$ This MO post might be helpful. $\endgroup$ – JHF May 8 at 15:39
  • $\begingroup$ 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). $\endgroup$ – peter a g May 8 at 17:40
  • $\begingroup$ 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 $\endgroup$ – peter a g May 8 at 18:00

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