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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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Yes, this is stated e.g. on page 3 of Hovey: Model category structures on chain complexes of sheaves.

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  • $\begingroup$ what is the generators of Ch(Mod-R)? $\endgroup$
    – user41630
    Commented Oct 21, 2013 at 10:10
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    $\begingroup$ For generators of $\mathbf{Ch(Mod_R)}$, see page 3 of the link provided in this answer. $\endgroup$
    – Todd Trimble
    Commented Oct 21, 2013 at 13:12

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