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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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See also the nlab page: ncatlab.org/nlab/show/… –  Adeel Oct 21 '13 at 16:23
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up vote 5 down vote accepted

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

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