  In this work we study the failure of the HKR theorem over rings of positive
and mixed characteristic. For this we construct a filtered circle interpolating
between the usual topological circle and a formal version of it. By mapping to
schemes we produce this way an interpolation, realized in practice by the
existence of a natural filtration, from Hochschild and (a filtered version of)
cyclic homology to derived de Rham cohomology. In particular, we show that this
recovers the filtration of Antieau and Bhatt-Morrow-Scholze. The construction
of our filtered circle is based on the theory of affine stacks and affinization
introduced by the third author, together with some facts about schemes of Witt
vectors.
