  Following an old suggestion of M. Kontsevich, and inspired by recent work of
A. Beilinson and B. Bhatt, we introduce a new version of periodic cyclic
homology for DG agebras and DG categories. We call it co-periodic cyclic
homology. It is always torsion, so that it vanishes in char 0. However, we show
that co-periodic cyclic homology is derived-Morita invariant, and that it
coincides with the usual periodic cyclic homology for smooth cohomologically
bounded DG algebras over a torsion ring. For DG categories over a field of odd
positive characteristic, we also establish a non-commutative generalization of
the conjugate spectral sequence converging to our co-periodic cyclic homology
groups.
