Periodic cyclic homology over ℚ

Let X be a derived scheme over an animated commutative ring of characteristic 0. We give a complete description of the periodic cyclic homology of X in terms of the Hodge completed derived de Rham complex of X.In particular this extends earlier computations of Loday-Quillen to non-smooth algebras.Moreover, we get an explicit condition on the Hodge completed derived de Rham complex, that makes the HKR-filtration on periodic cyclic homology constructed by Antieau and Bhatt-Lurie exhaustive.