Given an abelian category $\mathcal{A}$ the category of chain complexes over $\mathcal{A}$ is again an abelian category. If $\mathcal{A}$ is a Grothendieck category then the category of chain complexes over $\mathcal{A}$ is a Grothendieck category? In praticular, for a ring $R$ with unitary and the category of its left unitary modules, does this hold? 


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About the category of chain complexes and Grothendieck categories.
Yes, this is stated e.g. on page 3 of [Hovey: Model category structures on chain complexes of sheaves](http://www.math.uiuc.edu/K-theory/0366/sheaves.pdf).


