Derived category of filtered objects

Abstract

For an abelian category C and a filtrant preordered set Λ, we prove that the derived category of the quasi-abelian category of filtered objects in C indexed by Λ is equivalent to the derived category of the abelian category of functors from Λ to C . We apply this result to the study of the category of filtered modules over a filtered ring in a tensor category.

Construction of sheaves on the subanalytic site

“…For T = M, M sa , M sal , the category Mod(FD T ) of filtered D-modules on T is quasi-abelian in the sense of [Sch99] and its derived category D + (FD T ) is well-defined. We shall use here the recent results of [SS16] which give an easy description of these derived categories and we construct a right adjoint ρ ! sal to the derived functor Rρ sal * : D + (FD Msa ) − → D + (FD M sal ).…”