Derived category of filtered objects

TL;DR

This paper establishes an equivalence between the derived category of filtered objects in an abelian category and the derived category of functors from the indexing set, with applications to filtered modules over filtered rings.

Contribution

It proves a new equivalence of derived categories for filtered objects and functors, extending understanding in filtered module categories within tensor categories.

Findings

Derived category of filtered objects is equivalent to functor category

Application to filtered modules over filtered rings

Provides tools for studying filtered objects in tensor categories

Abstract

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