INTRODUCTION 5

However, the definition of coderived categories in [37, 30] was not entirely satis-

factory, in our view, in that the right hand side of the purported duality is to a

certain extent defined in terms of the left hand side (the approach to D-Ω duality

developed in [5, section 7.2] had the same problem). This defect is corrected in the

present paper.

The analogous problem is present in the definitions of model category struc-

tures on the categories of DG-coalgebras by Hinich [22] and Lef` evre-Hasegawa [37].

This proved harder to do away with: we obtain various explicit descriptions of the

distinguished classes of morphisms of CDG-coalgebras independent of the Koszul

duality, but the duality functors are still used in the proofs.

In addition, we emphasize contramodules and CDG-coalgebras, whose role in

the derived categories of the second kind and derived Koszul duality business does

not seem to have been appreciated enough.

0.9. Now let us describe the content of this paper in more detail. In Sec-

tion 1 we obtain two semiorthogonal decompositions of the homotopy category of

DG-modules over a DG-ring, providing injective and projective resolutions for the

derived category of DG-modules. We also consider flat resolutions and use them to

define the derived functor Tor for a DG-ring. Besides, we construct a t-structure

on the derived category of DG-modules over an arbitrary DG-ring. In anticipa-

tion of the forthcoming paper [49], we also discuss silly filtrations on the same

derived category. This section contains no new results; it is included for the sake

of completeness of the exposition.

The derived categories of DG-comodules and DG-contramodules and the dif-

ferential derived functors

CotorC,I

and CoextC

I

of the first kind for a DG-coalgebra

C are briefly discussed in Section 2, the proofs of the main results of this section

being postponed to Sections 5 and 7. Partial results about injective and projective

resolutions for the coderived and contraderived categories of a CDG-ring are ob-

tained in Section 3. The finite homological dimension, Noetherian, coherent, and

Gorenstein cases are considered; in the former situation, a natural definition of the

differential derived functor

TorB,II

of the second kind for a CDG-ring B is given.

In addition, we construct an “almost involution” on the category of DG-categories.

In Section 4 we construct semiorthogonal decompositions of the homotopy cat-

egories of CDG-comodules and CDG-contramodules over a CDG-coalgebra, pro-

viding injective and projective resolutions for the coderived category of CDG-

comodules and the contraderived category of CDG-contramodules. We also define

the differential derived functors Cotor, Coext, and Ctrtor for a CDG-coalgebra,

and give a suﬃcient condition for a morphism of CDG-coalgebras to induce equiva-

lences of the coderived and contraderived categories. The comodule-contramodule

correspondence for a CDG-coalgebra is obtained in Section 5.

Koszul duality (or “triality”, as there are actually two module categories on

the coalgebra side) is studied in Section 6. Two versions of the duality theorem

for (C)DG-modules, CDG-comodules, and CDG-contramodules are obtained, one

valid for conilpotent CDG-coalgebras only and one applicable in the general case.

We also construct an equivalence between natural localizations of the categories of

DG-algebras (with nonzero units) and conilpotent CDG-coalgebras.

We discuss the derived categories of A∞-modules and the co/contraderived

categories of curved A∞-co/contramodules in Section 7. We explain the relation

between strictly unital A∞-algebras and coaugmented CDG-coalgebra structures