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Jan 28, 2019 at 16:05 comment added Leonid Positselski 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.
Jan 28, 2019 at 16:03 comment added Leonid Positselski 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.
Jan 28, 2019 at 13:01 history asked user108998 CC BY-SA 4.0