Periodic cyclic homology and derived de Rham cohomology

Abstract

We use the Beilinson $t$-structure on filtered complexes and the Hochschild–Kostant–Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt–Morrow–Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.