# Is the category of chain complexes over an ring $R$ a locally presentable category?

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.

• Do you know how to prove the category of $R$-modules is locally presentable? – Eric Wofsey Jul 2 at 6:12
• @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. – Dion Grand Jul 2 at 6:25