Two Kinds of Derived Categories, Koszul Duality, and by Leonid Positselski

By Leonid Positselski

The purpose of this paper is to build the derived nonhomogeneous Koszul duality. the writer considers the derived different types of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived different types of CDG-modules, the coderived type of CDG-comodules, and the contraderived class of CDG-contramodules. The equivalence among the latter different types (the comodule-contramodule correspondence) is validated. Nonhomogeneous Koszul duality or "triality" (an equivalence among unique derived different types such as Koszul twin (C)DG-algebra and CDG-coalgebra) is acquired within the conilpotent and nonconilpotent models. quite a few A-infinity constructions are thought of, and a few version class constructions are defined. Homogeneous Koszul duality and D-$\Omega$ duality are mentioned within the appendices.

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Proof. It suffices to show that any morphism L −→ M in Hot(B–mod) between objects L ∈ Hot(B–modfg ) and M ∈ Acyclco (B–modfg ) factorizes through an object of Acyclabs (B–modfg ). Indeed, any left CDG-module over B that can be obtained from total CDG-modules of exact triples of CDG-modules over B using the operations of cone and infinite direct sum is a filtered inductive limit of its CDGsubmodules that can be obtained from the total CDG-modules of exact triples of finitely generated CDG-modules using the operation of cone.

Then σ n M is a DG-submodule of M and the quotient DG-modules σ n−1 M/σ n M are contractible for any acyclic DG-module M over A. 30 LEONID POSITSELSKI Remark. The assertion (b) of Theorem 2 holds under somewhat weaker assumptions: the condition that A1 = 0 can be replaced with the condition that d(A0 ) = 0. The proof remains the same, except that the quotient DG-modules σ n−1 M/σ n M no longer have to be contractible when M is acyclic. 5. Semiorthogonality. Let B be a CDG-ring. Denote by Hot(B–modinj ) the full subcategory of the homotopy category of left CDG-modules over B formed by all the CDG-modules M for which the graded module M # over the graded ring B # is injective.

Let M ′ be any CDG-submodule in J properly containing M such that the quotient CDG-module M ′ /M is finitely generated. Since J # is injective, the graded B # -module morphism h : M −→ J of degree −1 can be extended to a graded B # -module morphism h′′ : M ′ −→ J of the same degree. Let ι : M −→ J and ι′ : M ′ −→ J denote the identity embeddings. The map ι′ − d(h′′ ) is a closed morphism of CDG-modules M ′ −→ J vanishing in the restriction to M , so it induces a closed morphism of # 3. CODERIVED AND CONTRADERIVED CATEGORIES OF CDG-MODULES 39 CDG-modules f : M ′ /M −→ J.

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