[Paper Review] Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

Leonid Positselski|May 17, 2009

Algebraic structures and combinatorial modelsMathematics35 references119 citations

TL;DR

This paper establishes a foundational framework for two distinct derived categories—derived categories of the first kind (standard derived categories) and the second kind (coderived and contraderived categories)—in the context of DG-modules, comodules, and contramodules over DG- and CDG-algebras. It proves that Koszul duality and the comodule-contramodule correspondence are unified through these derived categories, with key results showing equivalence of derived, coderived, and contraderived categories for cofibrant DG-algebras and the validity of the correspondence under finite homological dimension.

ABSTRACT

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived categories of CDG-comodules, and the contraderived categories of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or "triality" (an equivalence between exotic derived categories corresponding to Koszul dual (C)DG-algebra and CDG-coalgebra) is obtained in the conilpotent and nonconilpotent versions. Various $A_\infty$-structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and $D$-$Ω$ duality are discussed in the appendices.

Motivation & Objective

• To resolve convergence issues in spectral sequences arising in Koszul duality by distinguishing between two kinds of derived categories.
• To establish a duality between comodules and contramodules via coderived and contraderived categories.
• To generalize Koszul duality to nonconilpotent and curved settings using CDG-structures and A∞-algebras.
• To unify derived categories of DG-modules, comodules, and contramodules through model category and homotopical algebra techniques.
• To prove that coderived and contraderived categories coincide for CDG-rings with finite homological dimension, especially in the case of free underlying graded algebras.

Proposed method

• Introduces two derived categories: the standard derived category (quotient by acyclic complexes) and the coderived/contraderived category (quotient by complexes with injective/projective underlying modules without differential).
• Uses completion and filtration techniques to ensure spectral sequence convergence, particularly by replacing incomplete filtrations with complete ones.
• Applies the comodule-contramodule correspondence via a duality functor that identifies coderived and contraderived categories under finite homological dimension.
• Employs model category structures on DG-modules, CDG-comodules, and CDG-contramodules to construct homotopy categories and derive derived functors.
• Applies the ${ m D}$–$ ext{Hom}$ duality and Rees algebra techniques to show generation of bounded coherent derived categories by filtered modules.
• Uses the forgetful functor and totalization to relate absolute derived categories of CDG-modules to bounded derived categories of filtered modules.

Experimental results

Research questions

• RQ1 How can spectral sequences in Koszul duality be made to converge when standard filtrations fail due to incompleteness?
• RQ2 What is the precise relationship between coderived and contraderived categories in the context of DG- and CDG-structures?
• RQ3 In what cases do the derived, coderived, and contraderived categories of DG-modules over a DG-algebra coincide?
• RQ4 How does the comodule-contramodule correspondence extend to curved A∞-coalgebras and CDG-algebras?
• RQ5 Under what conditions does the derived category of a CDG-ring coincide with its coderived and contraderived counterparts?

Key findings

• Spectral sequences converge when using complete filtrations, and the two possible completions (infinite products vs. infinite direct sums) lead to two distinct total complexes.
• The derived category of the first kind is the quotient of the homotopy category by acyclic complexes, while the derived category of the second kind is equivalent to the full subcategory of complexes that are injective or projective when the differential is forgotten.
• For any DG-algebra, the derived, coderived, and contraderived categories of DG-modules are equivalent, as any DG-algebra is quasi-isomorphic to a cofibrant one.
• When the underlying graded algebra of a CDG-algebra is free, the coderived and contraderived categories coincide, and the comodule-contramodule correspondence holds.
• The absolute derived category of ${ m O}_X$-coherent CDG-modules is generated by filtered modules and is closed under idempotent completion, ensuring duality with bounded coherent categories.
• The Rees algebra of a filtered ${ m D}$-module is Noetherian and of finite homological dimension, allowing resolution of coherent filtered modules by sums of standard filtered modules.