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.