Given a pair of Koszul dual algebras, say $S^*(V)$ and $\bigwedge^*(V^*)$ for some vector space $V$, one obtains a triangulated equivalence between their bounded derived categories of finitely-generated graded modules.

Given a pair of Koszul dual operads, say the Lie and commutative operads, what is the precise analogue of a derived equivalence between their categories of algebras?


The situation for graded modules over a pair of Koszul dual algebras is more complicated, actually. What the question says is true for Koszul algebras $A$ and $A^!$ provided that $A$ is Noetherian and $A^!$ is finite-dimensional (including the case of the symmetric and exterior algebras) but not otherwise. In general one can say that the unbounded derived categories of positively graded modules with finite-dimensional components over $A$ and $A^!$ are anti-equivalent. The subcategories of complexes of positively graded modules bounded separately in every grading in these unbounded derived categories are also anti-equivalent.

One can replace the contravariant anti-equivalence with a covariant equivalence by considering positively graded modules over one of the algebras and negatively graded modules over the other one (both algebras being considered as positively graded). In this case one does not have to require the components of the modules to be finite-dimensional.

With algebras over operads, the analogue of the equivalence for graded modules involves DG-algebras with an additional positive grading (there being only the ground field $k$ in the additional grading $0$ and nothing in the negative additional grading), with the additional grading preserved by the differential. The Koszul duality is an anti-equivalence between the localizations of the categories of DG-algebras of this kind, with every component of fixed additional grading being a bounded complex of finite-dimensional vector spaces, by quasi-isomorphisms. For some operads (e.g., for Lie and Com) one has to assume the field $k$ to have characteristic $0$, while for some others (e.g., Ass) one doesn't.

If one wishes to replace the contravariant anti-equivalence with a covariant equivalence in the case of algebras over operads, one has to consider algebras on one side of the equivalence and coalgebras on the other side. Then the boundedness and finite-dimensionality requirements can be dropped.

What I've described above is the homogeneous Koszul duality; the nonhomogeneous case (with ungraded modules or algebras without the additional grading) is more complicated, though also possible. See my answer to the question linked to from the question above.

References: 1. Beilinson, Ginzburg, Soergel "Koszul duality patterns in representation theory", 2. My preprint "Two kinds of derived categories, ...", arXiv:0905.2621, Appendix A.


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