Enhancing the filtered derived category

TL;DR

This paper generalizes the concept of filtered derived categories using advanced homotopical algebra tools, enabling new constructions and interactions with spectral sequences and monoidal structures.

Contribution

It introduces a natural framework for filtered stable infinity-categories and explores their properties and applications in homotopical algebra and related fields.

Findings

Characterization of filtered stable infinity-categories as localizations

Construction of a model category of filtered D-modules

Development of filtered operads and algebras over operads

Abstract

The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical algebra, in the formalisms of stable infinity-categories, stable model categories, and pretriangulated, idempotent-complete dg categories. We characterize the filtered stable infinity-category Fil(C) of a stable infinity-category C as the left exact localization of sequences in C along the infinity-categorical version of completion (and prove analogous model and dg category statements). We also spell out how these constructions interact with spectral sequences and monoidal structures. As examples of this machinery, we construct a stable model category of filtered D-modules and develop the rudiments of a theory of filtered operads and filtered algebras over operads.