# Is the Quillen Injective Model Structure on the category of positive cochain complexes of R-modules (co)fibrantly generated?

Denoted with $Ch^+_R$ the category of positive cochain complexes of R-modules (for a commutative ring $R$), it admits a model structure where:

• weak-equivalences are quasi-isomorphisms;
• cofibrations are degreewise monomorphisms;
• fibrations are degreewise epimorphisms with injective kernels.

This structure is known as Quillen injective model structure on $Ch_R^+$.

Is this model structure cofibrantly generated (according to the definition given by M. Hovey in the second chapter of his book "Model Categories")? And if so, which are its generating cofibration and trivial cofibrations? I'm pretty sure the answer is positive, but I can't figure out why (I'm quite new to these concepts and I can't find a proper reference).

EDIT: The Quillen projective model structure on the category of positive chain complexes of modules, where:

• weak-equivalences are quasi-isomorphisms;
• cofibrations are degreewise monomorphisms with projective kernels;
• fibrations are degreewise epimorphisms;

should be cofibrantly generated, with the classes $I$ and $J$ of generating cofibration and trivial cofibrations defined as follows: for every integer $n$ we define $S_n\in Ch_R$ to be a complex with $S_n^n=M$ and $0$ otherwise (endowed with a trivial differential) and $D_n$ to be a complex with $D^k_n$ for $k=n-1,n$ and $0$ otherwise (its differential is the identity in degree $n$ and is trivial otherwise). Then $I$ is the class of natural inclusions $S_n\hookrightarrow D_n$ and $J$ is the class of natural inclusions $0\hookrightarrow D_n$.

In this article is proved that $J$ is a set of generating trivial cofibrations for this structure (Lemma 2.5.3). Moreover Hovey shows in his book "Model Categories" that $I$ is a set of generating trivial cofibrations for his model structure on the category of unbounded chain complexes (whose fibrations are still degree-wise epimorphisms). I think it could work also for the Quillen projective structure.

Is it possible that the injective model structure is FIBRANTLY generated by the sets of maps $J'=\left\{D^n\twoheadrightarrow 0\right\}$ and $I'=\left\{D^n\twoheadrightarrow S^n\right\}$?

• I do not have a clue. – marty cohen Aug 16 '15 at 18:22
• I think the answer is yes, but it may be difficult to get an explicit generating set of trivial cofibrations. The key phrase is "Smith's theorem". – Zhen Lin Aug 16 '15 at 23:27
• In Theorem 2.3.13 Hovey describes the injective model structure for unbounded complexes and states that it is cofibrantly generated. The dual of Lemma 2.3.6 gives 'fibrant' = 'injective' for bounded above complexes. But you look at bounded below complexes and I am not sure what happens there. – Marc Olschok Aug 17 '15 at 1:12
• Thanks, I guess I didn't read that part very carefully. I'm going to update the question. – LK512 Aug 17 '15 at 9:35