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.

arXiv comment46 pages. Comments and questions are very welcome. v2: Improved the section on duals. v3: Added a section on dg categories

References in corpus (17)

• Higher Topos Theory
• Algebras and modules in monoidal model categories
• Uniqueness of enhancement for triangulated categories
• On the algebraic K-theory of higher categories
• The pro-étale topology for schemes
• Admissibility and rectification of colored symmetric operads
• Completions and derived de Rham cohomology
• Differential Graded Categories are k-linear Stable Infinity Categories
• On (Enriched) Left Bousfield Localization of Model Categories
• Enriched model categories and presheaf categories
• Simplicial nerve of an A-infinity category
• Homotopy theory of symmetric powers
• Homology of moduli spaces of curves and commutative homotopy algebras
• Day convolution for infinity-categories
• t-structures are normal torsion theories
• Derived category of filtered objects
• 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