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 . 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 , 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 , 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 =
The graded S-module M = k[x,
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