# model structuctures on the category of unbounded chain complexes

In his book "Model Categories" Mark Hovey constructs both projective and injective model structures on unbounded chain complexes of $R$-modules. For what kind of abelian categories this construction works? For example what about the categorty $I^{R-mod}$ of diagrams of $R$-modules of a fixed shape $I$?

A version of this question was already asked, but there the emphasis was on the category of sheaves, and I am interested in a category of diagrams that always has enough projectives and injectives. Any reference will be appreciated.

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Here's a paper where Hovey discusses model sructures on unbounded chain complexes in an abelian category: arxiv.org/abs/math/0011216. – Marc Hoyois Sep 20 '12 at 21:35

In your case, $R$-mod has two complete cotorsion pairs: $(C, W\cap F)$ and $(C\cap W, F)$, by Prop 2.2. in the linked paper. You can promote those to pairs on $I^{R-mod}$ via $(I^C, I^{W\cap F})$ and $(I^{C\cap W}, I^F)$ using the fact that Ext groups between diagrams should be computed object-wise (I can't imagine how else you would compute them). It's also easy to see that a diagram of elements of the form $W\cap F$ is an intersection of diagrams from $I^W$ and $I^F$, so you can write $I^{W\cap F}$ as $I^W \cap I^F$. You say in your question that the diagrams have enough projectives and injectives, so this means you get complete cotorsion pairs. All you have to do to use Theorem 2.5 of the paper and hence get an abelian model structure is to show $I^W$ is thick, i.e. closed under retracts and satisfies the 2-out-of-3 property. Both should be easy, since $W$ is thick.