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.