6 LEONID POSITSELSKI
on graded tensor coalgebras, and use it to prove the standard results about strictly
unital A∞-modules. The similar approach to strictly counital curved A∞-coalgebras
yields the comodule-contramodule correspondence in the A∞ case.
Model category structures (of the first kind) for DG-modules over a DG-ring
and model category structures (of the second kind) for CDG-comodules and CDG-
contramodules over a CDG-coalgebra are constructed in Section 8. We also ob-
tain model category structures of the first kind for DG-comodules and DG-contra-
modules over a DG-coalgebra, and model category structures of the second kind for
CDG-modules over a CDG-ring in the finite homological dimension, Noetherian,
coherent, and Gorenstein cases. Quillen equivalences related to the comodule-
contramodule correspondence and Koszul duality are discussed.
We consider the model categories of DG-algebras and conilpotent CDG-co-
algebras in Section 9. More precisely, it turns out that the latter category has to be
“finalized” in order to make it a model category. We also discuss DG-modules over
cofibrant DG-algebras. Conilpotent curved A∞-coalgebras and co/contranilpotent
curved A∞-co/contramodules over them are introduced.
Homogeneous Koszul duality is worked out in Appendix A. The (more general)
covariant and the (more symmetric) contravariant versions of the duality are con-
sidered separately. The equivalence between the derived category of modules over
the ring/sheaf of differential operations acting in a vector bundle and the coderived
category of CDG-modules over the corresponding de Rham CDG-algebra is con-
structed in Appendix B. A desription of the bounded derived category of coherent
D-modules in terms of coherent CDG-modules is also obtained.
Acknowledgement. The author is grateful to Michael Finkelberg for posing
the problem of constructing derived nonhomogeneous Koszul duality. I want to
express my gratitude to Vladimir Voevodsky for very stimulating discussions and
encouragement, without which this work would probably never have been done. I
also benefited from discussions with Joseph Bernstein, Victor Ginzburg, Amnon
Neeman, Alexander Beilinson, Henning Krause, Maxim Kontsevich, Tony Pantev,
Alexander Polishchuk, Alexander Kuznetsov, Lars W. Christensen, Jan
ˇˇ
S tov´ıˇ cek,
Pedro Nicol´ as, and Alexander Efimov. I am grateful to Ivan Mirkovic, who always
urged me to write down the material presented below. I want to thank Dmitry
Arinkin, who communicated the proof of Theorem 3.11.2 to me and gave me the
permission to include it in this paper. Most of the content of this paper was worked
out when I was a Member of the Institute for Advanced Study, which I wish to thank
for its hospitality. I am also indebted to the participants of an informal seminar
at IAS, where I first presented these results in the Spring of 1999. The author
was partially supported by P. Deligne 2004 Balzan prize, an INTAS grant, and an
RFBR grant while writing the paper up. Parts of this paper have been written
when I was visiting the Institut des Hautes
´
Etudes Scientifiques, which I wish to
thank for the excellent working conditions.
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