INTRODUCTION 3

0.4. Now let us turn to (derived) Koszul duality. This subject originates from

the classical Bernstein–Gelfand–Gelfand duality (equivalence) between the bounded

derived categories of finitely generated graded modules over the symmetric and

exterior algebras with dual vector spaces of generators [8]. Attempting to generalize

this straightforwardly to arbitrary algebras, one discovers that many restricting

conditions have to be imposed: it is important here that one works with algebras

over a field, that the algebras and modules are graded, that the algebras are Koszul,

that one of them is finite-dimensional, while the other is Noetherian (or at least

coherent) and has a finite homological dimension.

The standard contemporary source is [7], where many of these restrictions are

eliminated, but it is still assumed that everything happens over a semisimple base

ring, that the algebras and modules are graded, and that the algebras are Koszul.

In [6], Koszulity is not assumed, but positive grading and semisimplicity of the

base ring still are. The main goal of this paper is to work out the Koszul duality

for ungraded algebras and coalgebras over a field, and more generally, differential

graded algebras and coalgebras. In this setting, the Koszulity condition is less

important, although it allows to obtain certain generalizations of the duality results.

As to the duality over a base more general than a field, in this paper we only

consider the special case of D–Ω duality, i. e., the duality between complexes of

modules over the ring of differential operators and (C)DG-modules over the de

Rham (C)DG-algebra of differential forms (see 0.6). The ring of functions (or

sections of the bundle of endomorphisms of a vector bundle) is the base ring in this

case. For a more general treatment of the relative situation, we refer the reader

to [48, Chapter 11], where a version of Koszul duality is obtained for a base coring

over a base ring.

The thematic example of nonhomogeneous Koszul duality over a field is the

relation between complexes of modules over a Lie algebra g and DG-comodules over

its standard homological complex. Here one discovers that, when g is reductive, the

standard homological complex with coeﬃcients in a nontrivial irreducible g-module

has zero cohomology—even though it is not contractible, and becomes an injective

graded comodule when one forgets the differential. So one has to consider a version

of derived category of DG-comodules where certain acyclic DG-comodules survive if

one wishes this category to be equivalent to the derived category of g-modules. That

is how derived categories of the second kind appear in Koszul duality [17, 37, 30].

0.5. Let us say a few words about the homogeneous case. In the generality

of DG-(co)algebras, the homogeneous situation is distinguished by the presence of

an additional positive grading preserved by the differentials. Such a grading is

well-known to force convergence of the spectral sequences, so there is no difference

between the derived categories of the first and the second kind in the homogeneous

case. It is very essential here that the grading be indeed positive (or negative) on

the DG-(co/contra)modules as well as the DG-(co)algebras, as one can see already

in the example of the duality between the symmetric and the exterior algebras in

one variable, S = k[x] and Λ =

k[ε]/ε2.

The graded S-module M = k[x,

x−1]

corresponds to the acyclic complex of Λ-modules K = (··· −→ Λ −→ Λ −→ · · · )

whose every term is Λ and every differential is the multiplication with ε.

The acyclic, but not contractible complex K of projective and injective Λ-

modules provides the simplest way to distinguish between the derived categories of

the first and the second kind. In derived categories of the first kind, it represents