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 classification of CDG-comodules, and the contraderived classification of CDG-contramodules. The equivalence among the latter different types (the comodule-contramodule correspondence) is proven. Nonhomogeneous Koszul duality or "triality" (an equivalence among unique derived different types akin to Koszul twin (C)DG-algebra and CDG-coalgebra) is got within the conilpotent and nonconilpotent types. numerous A-infinity buildings are thought of, and a few version classification constructions are defined. Homogeneous Koszul duality and D-$\Omega$ duality are mentioned within the appendices.
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Additional info for Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence
There is also an exact triple of graded B # -modules L −→ G+ (L)# −→ L[−1]. So, in particular, we have a closed surjective morphism of CDG-modules G+ (P0 ) −→ M , where the graded B # -module G+ (P0 )# is projective. Let K be the kernel of the surjective morphism G+ (P0 ) −→ M (taken in the abelian category Z 0 DG(B–mod) of CDG-modules and closed morphisms between them). Applying the same procedure to the CDG-module K in place of M , we obtain the CDG-module G+ (P1 ), etc. Since the graded left homological dimension of B # is ﬁnite, there exists a nonnegative integer d such that the image Z of the morphism G+ (Pd ) −→ G+ (Pd−1 ) taken in the abelian category Z 0 DG(B–mod) is projective as a graded B-module.
CDG-rings and CDG-modules. A CDG-ring (curved diﬀerential graded ring) B = (B, d, h) is a triple consisting of an associative graded ring B = i∈Z B i , an odd derivation d : B −→ B of degree 1, and an element h ∈ B 2 satisfying the equations d2 (x) = [h, x] for all x ∈ B and d(h) = 0. A morphism of CDG-rings f : B −→ A is a pair f = (f, a) consisting of a morphism of graded rings f : B −→ A and an element a ∈ A1 satisfying the equations f (dB (x)) = dA (f (x)) + [a, x] and f (hB ) = hA + dA (a) + a2 for all x ∈ B, where B = (B, dB , hB ) and A = (A, dA , hA ), while the bracket [y, z] denotes the supercommutator of y and z.
This derived functor coincides with the above-deﬁned functor TorB,II , since one can use ﬁnite resolutions P• and Q• in the construction of the latter functor in the ﬁnite weak homological dimension case. Analogously, whenever the graded ring B # has a ﬁnite left homological dimension, the functor HomDabs (B–mod) (L, M ) of homomorphisms in the absolute derived category coincides with the above-deﬁned functor ExtII B (L, M ). 4. 1. CDG-comodules and CDG-contramodules. Let k be a ﬁxed ground ﬁeld.
Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence by Leonid Positselski