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## Co-periodic cyclic homology

Advances in Mathematics. 2018. Vol. 334. P. 81-150.

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 algebras 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.