7
$\begingroup$

The following might be very well known for people who works with model categories, but I do not find the answer.

Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain complexes of $A$-modules (complexes where differential rises the degree and which are zero in negative degree). There is the injective model structure on $\mathbf{Ch}_+(A)$ which has quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations and epimorphisms with injective kernel as fibrations. Let $K^\bullet$ be a complex, then its injective resolution $I^\bullet (K^\bullet)$ turn out to be its fibrant replacement.

Let $X$ be a topological space and consider $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$, the category of positive complexes of sheaves of $A$-modules on $X$. My question is the following:

Question: Is there an analogue model structure in $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$ to the injective model structure in $\mathbf{Ch}_+(A)$? More concretely, if we consider quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations, and morphisms having the right lifting property with respect to injective quasi-isomorphisms, do they define a model structure? If $\mathcal{K^\bullet}$ is a complex of sheaves, will its injective resolution $I^\bullet (\mathcal{K}^\bullet)$ be its fibrant replacement?

Thank you very much

$\endgroup$
  • 1
    $\begingroup$ I don't have time to write a full answer just now, but I think the answer is probably yes, and that proof should use cotorsion pairs. I recommend you look at the work of Hovey and Gillespie. Gillespie has a paper handling categories of chain complexes of quasi-coherent sheaves over a scheme, and perhaps some mention is made there of the case you care about. I believe there are two cotorsion pairs in these settings, one for the projective model structure and one for the injective. $\endgroup$ – David White Aug 30 '15 at 22:22
  • $\begingroup$ I think that, in all cases, cofibrations should be cochain maps which are mono in strictly positive degrees (not necessarily in degree $0$), so that the $t$-structure truncation functor from the unbounded category is a left Quillen functor. $\endgroup$ – Fernando Muro Aug 31 '15 at 0:03
4
$\begingroup$

David White's comment led me to adequate references to answer this question. Thank you very much

Question: Is there an analogue model structure in $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$ to the injective model structure on $\mathbf{Ch}_+(A)$? More concretely, if we consider quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations, and morphisms having the right lifting property with respect to injective quasi-isomorphisms, do they define a model structure?

Answer: Yes, there is such model structure. The above mentioned three clasess do form a model structure. Find the result in a letter to A. Grothendieck from Joyal in Théorème 2 (p.10)

Question: If $\mathcal{K}^\bullet$ is a complex of sheaves, will its injective resolution $I^\bullet(\mathcal{K}^\bullet)$ be its fibrant replacement?

Answer: Yes, it will. The same argument that shows that for a chain complex of modules its fibrant replacement is a complex made of injective modules holds for sheaves. Find the argument for modules (although the case of projective resolutins) in this paper,

$\endgroup$
  • 1
    $\begingroup$ Great! Glad you got it resolved! $\endgroup$ – David White Sep 6 '15 at 15:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.