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 [48], as indeed, its key ideas precede those of [48] both historically and
logically. It would had been all but impossible to invent the use of exotic derived
categories for the purposes of [48] if these were not previously discovered in the
work presented below. Nevertheless, most results of this paper are not covered
by [48], since it is written in the generality of DG- and CDG-modules, comodules,
and contramodules, while [48] deals with nondifferential semi(contra)modules most
of the time. This paper is focused on Koszul duality, while the goal of [48] is the
semi-infinite cohomology.
The fact that exotic derived categories arise in Koszul duality was essentially
discovered by Hinich [22], whose ideas were developed by Lef` evre-Hasegawa [37,
Chapitres 1 and 2]; see also Fløystad [17], Huebschmann [24], and Nicol´ as [43]. The
terminology of “coderived categories” was introduced in Keller’s exposition [30].
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