2
$\begingroup$

Positselski's work allows one to frame Koszul duality very elegantly in terms of so called coderived categories of modules over coalgebras, these are somewhat exotic dg categories of comodules over $C$ in which certain acyclic complexes survive.

The reader can consult https://arxiv.org/abs/0905.2621 for a detailed exposition.

My question, if I have a coalgebra $C$, is there an explicit, and ideally "fairly small" complex, defined from $C$, which computes the Hochschild (co-) homology of the coderived category of comodules for $C$?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ A relevant reference is our paper A.Polishchuk, L.Positselski "Hochschild (co)homology of the second kind I", Trans. Amer. Math. Soc. 364 (2012), arxiv.org/abs/1010.0982 , in which Hochschild homology and cohomology theories loosely associated with derived categories of the second kind are defined for CDG-algebras and CDG-modules. $\endgroup$
    – Leonid Positselski
    Commented Jan 28, 2019 at 16:03
  • 3
    $\begingroup$ Hochschild homology and perhaps also cohomology of the second kind for a CDG-coalgebra would be an even more natural and better behaved construction, closely related to the coderived categories of CDG-comodules (and perhaps also the contraderived categories of CDG-contramodules). This looks quite doable, but it appears that noone has done it yet. $\endgroup$
    – Leonid Positselski
    Commented Jan 28, 2019 at 16:05

0

Reset to default

You must log in to answer this question.