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Given a ringed space $(X,\mathcal{O})$ of can construct the flat model structure on chain complexes of $\mathcal{O}$-modules:

  • Weak equivalences are quasi-isomorphisms
  • The fibrations are epimorphisms with dg-cotorsion kernels
  • The cofibrations are monomorphisms with dg-flat cokernels

Gillespie (2006) constructed this and showed that it is monoidal, i.e. the derived tensor product can be computed via cofibrant resolutions.

Does this model category structure also make the pullback / pushforward adjunction into a Quillen adjunction? More precisely, given a morphism of ringed spaces $f:(X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ is $f^*\dashv f_*$ a Quillen adjunction?

If not in general, what conditions are needed on $f$, $X$, or $Y$?

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