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 class of CDG-comodules, and the contraderived class of CDG-contramodules. The equivalence among the latter different types (the comodule-contramodule correspondence) is demonstrated. Nonhomogeneous Koszul duality or "triality" (an equivalence among unique derived different types resembling Koszul twin (C)DG-algebra and CDG-coalgebra) is acquired within the conilpotent and nonconilpotent models. a variety of A-infinity buildings 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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Let J be a left CDG-module over B such that the graded B # -module J is injective. Suppose that the complex HomB (L, J) is acyclic for any ﬁnitely generated left CDG-module L over B. We have to prove that J is contractible. Apply Zorn’s Lemma to the ordered set of all pairs (M, h), where M is a CDGsubmodule in J and h : M −→ J is a contracting homolopy for the identity embedding M −→ J. It suﬃces to check that whenever M = J there exists a CDGsubmodule M ⊂ M ′ ⊂ J, M = M ′ and a contracting homotopy h′ : M ′ −→ J for the identity embedding M ′ −→ J that agrees with h on M .

Much more generally, to deﬁne the coderived (contraderived) category, it suﬃces to have a DG-category DG with shifts, cones, and arbitrary inﬁnite direct sums (products), for which the additive category Z 0 (DG) is endowed with an exact category structure. Examples of such a situation include not only the categories of CDG-modules, but also, e. , the category of complexes over an exact category [48]. Then one considers the total objects of exact triples in Z 0 (DG) as objects of the homotopy category H 0 (DG) and takes the quotient category of H 0 (DG) by the minimal triangulated subcategory containing all such objects and closed under inﬁnite direct sums (products).

The same observation allows to deduce (d) from (a). Theorem 2. Assume that Ai = 0 for all i < 0, the ring A0 is semisimple, and A = 0. Then (a) Acyclco,− (A–mod) = Acycl− (A–mod) and Acyclctr,+ (A–mod) = Acycl+ (A–mod); (b) the natural functors Hot± (A–mod)/Acycl± (A–mod) −→ D(A–mod) are fully faithful; (c) the natural functors Hot± (A–mod)/Acyclco,± (A–mod) −→ Dco (A–mod) and ± Hot (A–mod)/Acyclctr,± (A–mod) −→ Dctr (A–mod) are fully faithful; (d) the triangulated subcategories Acyclco (A–mod) and Acyclctr (A–mod) generate the triangulated subcategory Acycl(A–mod).