Invariance and localization for cyclic homology of DG algebras Original Research Article

We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [[19] and [4] preserves cyclic homology. This completes results of Rickardʹs [48] and Happelʹs [18]. It also extends the well-known results on preservation of cyclic homology under Morita equivalence [[10], [39], [25], [26], [41] and [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of Thomason-Trobaughʹs [51] and Yaoʹs [58] localization theorems in K-theory (cf. also [55]).