4 LEONID POSITSELSKI
the zero object and is not adjusted to various derived functors, while in derived
categories of the second kind, it is adjusted to derived functors and represents a
nonzero object. So the S-module M has to be excluded from the category of mod-
ules under consideration for a duality between conventional derived categories of
S-modules and Λ-modules to hold. The positivity condition on the internal grading
accomplishes that much in the homogeneous case. All the stronger conditions on
the gradings considered in [7, 6] are unnecessary for the purposes of establishing
derived Koszul duality.
0.6. Several attempts have been made in the literature [28, 5] to obtain an
equivalence between the derived category of modules over the ring/sheaf of dif-
ferential operators and an appropriately defined version of derived category of
DG-modules over the de Rham complex of differential forms. More generally, let
X be a smooth algebraic variety and E be a vector bundle on X with a global
connection ∇. Let Ω(X, End(E)) be the sheaf of graded algebras of differential
forms with coeﬃcients in the vector bundle End(E) of endomorphisms of E, d∇ be
the de Rham differential in Ω(X, End(E)) depending on the connection ∇, and
End(E)) be the curvature of ∇. Then the triple consisting of the sheaf
of graded rings Ω(X, End(E)), its derivation d∇, and the section h∇ is a sheaf of
CDG-rings over X. The derived category of modules over the sheaf of rings DX,E of
differential operators acting in the sections of the vector bundle E on X turns out to
be equivalent to the coderived category (and the contraderived category, when X is
aﬃne) of quasi-coherent CDG-modules over the sheaf of CDG-rings Ω(X, End(E)).
The assumption about the existence of a global connection in E can be dropped
(see subsection B.1 for details).
0.7. Yet another very good reason for considering derived categories of the
second kind is that in their terms a certain relation between comodules and con-
tramodules can be established. Namely, the coderived category of CDG-comodules
and the contraderived category of CDG-contramodules over a given CDG-coalgebra
are naturally equivalent. We call this phenomenon the comodule-contramodule cor-
respondence; it appears to be almost as important as the Koszul duality.
One can generalize the comodule-contramodule correspondence to the case of
strictly counital curved A∞-comodules and A∞-contramodules over a curved A∞-
coalgebra by considering the derived category of the second kind for CDG-modules
over a CDG-algebra whose underlying graded algebra is a free associative algebra.
0.8. This paper can be thought of as an extended introduction to the mono-
graph , as indeed, its key ideas precede those of  both historically and
logically. It would had been all but impossible to invent the use of exotic derived
categories for the purposes of  if these were not previously discovered in the
work presented below. Nevertheless, most results of this paper are not covered
by , since it is written in the generality of DG- and CDG-modules, comodules,
and contramodules, while  deals with nondifferential semi(contra)modules most
of the time. This paper is focused on Koszul duality, while the goal of  is the
The fact that exotic derived categories arise in Koszul duality was essentially
discovered by Hinich , whose ideas were developed by Lef` evre-Hasegawa [37,
Chapitres 1 and 2]; see also Fløystad , Huebschmann , and Nicol´ as . The
terminology of “coderived categories” was introduced in Keller’s exposition .