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rephrase the question
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Jakob
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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?

  1. For what rings does pseudo-coherence imply reflexivity in this derived sense?
  2. 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?

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 module 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?

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. Another 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.

  1. For what rings does pseudo-coherence imply reflexivity in this derived sense?
  2. 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?
integers
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Jakob
  • 2k
  • 12
  • 18

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 module 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?

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 module 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?), i.e., are reflexive modules (where reflexivity is always understood in the derived sense) pseudo-coherent?

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 module 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?

Source Link
Jakob
  • 2k
  • 12
  • 18

Reflexive vs. pseudo-coherent abelian groups

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 module 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?), i.e., are reflexive modules (where reflexivity is always understood in the derived sense) pseudo-coherent?