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As far as I remember this is explained (maybe in differet terms) in Kapranov's paper

"On DG-modules over the de rham complex and the vanishing cycles functor", Lecture Notes in Mathematics, 1991, Volume 1479.

You could also have a look at the discussion below this entry of the everything seminar.

I think the kind of statement you are expecting ("equivalences between carefully defined versions of their "derived categories") are of a type you can find in this paper.

In any way, the equivalence is given by a $\otimes$-Hom adjunction. I might have misunderstood something but I don't really understand where is the problem with having $Hom(\Omega,-)$ and $\Omega\otimes -$.

EDIT: more precisely, I have the feeling that the statement is the following.

If $A$ is Koszul and if $B$ is its Koszul dual then $Hom(A,-)$ and $A\otimes-$ defines an equivalence of DG categories between

• $B$-modules

• the category $\mathcal P_A$ as it is defined e.g. in Block's paper.

1

As far as I remember this is explained (maybe in differet terms) in Kapranov's paper

"On DG-modules over the de rham complex and the vanishing cycles functor", Lecture Notes in Mathematics, 1991, Volume 1479.

You could also have a look at the discussion below this entry of the everything seminar.

I think the kind of statement you are expecting ("equivalences between carefully defined versions of their "derived categories") are of a type you can find in this paper.

In any way, the equivalence is given by a $\otimes$-Hom adjunction. I might have misunderstood something but I don't really understand where is the problem with having $Hom(\Omega,-)$ and $\Omega\otimes -$.