Enhancing the filtered derived category

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.

46 pages. Comments and questions are very welcome. v2: Improved the section on duals. v3: Added a section on dg categories

References in corpus (5)

• Higher Topos Theory
• Completions and derived de Rham cohomology
• On (Enriched) Left Bousfield Localization of Model Categories
• Homotopy theory of symmetric powers
• Operadic comodules and (co)homology theories

Cited by in corpus (11)

• Topological Hochschild homology and integral $p$-adic Hodge theory
• Periodic cyclic homology and derived de Rham cohomology
• Deformation Theory and Partition Lie Algebras
• A Universal HKR Theorem
• A descent principle for compactly supported extensions of functors
• The global derived period map
• Homotopy in Exact Categories
• A $t$-structure on the $\infty$-category of mixed graded modules
• Topological Hochschild homology and Zeta-values
• Derived $F$-zips
• Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause