I wonder if the category of chain complexes over an ring $R$ is a locally presentable category.

I am trying proving that this category is combinatorial, I have seen some reference for the cofibrantly generated part, but for another part, I don't have any idea, is it really locally presentable?

So far I have considered the collection $C$ of all bounded chain complexes of finitely presented $R$-modules. However, I don't know how to prove that it's a small set and how to prove that all chain complexes are $\lambda$-filtered colimits of these chain complexes in $C$ for some regular cardinal $\lambda$.

Can someone give me some hint? Any ideas are welcome.

Thank you very much.

  • $\begingroup$ Do you know how to prove the category of $R$-modules is locally presentable? $\endgroup$ – Eric Wofsey Jul 2 at 6:12
  • $\begingroup$ @EricWofsey I don’t know. But I think it’s a particular case of my question since a $R$-module can be considered as a chain complex concentrated at 0. $\endgroup$ – Dion Grand Jul 2 at 6:25

Yes it is. It's one of the main examples other than simplicial sets.

Check T.Beke's paper " Sheafifiable homotopy model categories" Proposition 3.10

  • $\begingroup$ Thank you very much $\endgroup$ – Dion Grand Jul 18 at 11:03

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