Recall that a module M over some ring R is pseudo-coherent if it admits a resolution whose terms are finitely generated projective R-modules. Such a moduleAnother notion is to ask whether M is reflexive when regarded as an object in the derived category $D(Mod_R)$, i.e., $$M \to RHom(RHom(M, R), R))$$ is a quasi-isomorphism, where RHom denotes the derived functor of the usual homomorphisms of R-modules.
Does the converse hold? (Maybe for nice rings, even for the integers?), i.e., are reflexive modules (where reflexivity is always understood in the derived sense) pseudo-coherent?
- For what rings does pseudo-coherence imply reflexivity in this derived sense?
- When does the converse hold? (Maybe for nice rings, even for the integers?), i.e., are reflexive modules (where reflexivity is always understood in the derived sense) pseudo-coherent?