Invariance and localization for cyclic homology of DG algebras
It is well known that cyclic homology \(HC_*\) is insensitive to Morita equivalence, and there are many ways to account for this fact. More generally, can one give good criteria for when the cyclic homology of two algebras coincide? In another -- and in the reviewer's opinion, more important -- direction, there is a need to understand the localization properties of \(HC_*\) in analogy with the known localization properties of algebraic K-theory. This paper addresses these (and more) problems, and gives interesting solutions. The author considers cyclic homology of DG algebras, and shows that \(HC_*\) can be considered as a functor of a category \({\mathcal {ALG}}\) of DG algebras. A morphism \(A\to B\) in \({\mathcal {ALG}}\) is an isomorphism class in a full subcategory \(\text{rep}(A,B)\) in the derived category of \(A\)-\(B\)-bimodules. Even better: cyclic homology factors through \({\mathcal {ALG}}\to {\mathcal {ALG}}_0\) where the latter category has the same objects, but where the morphisms from \(A\) to \(B\) are elements in the Grothendieck groups \(K_0(\text{rep}(A,B))\). The author phrases this as the K-theoretic invariance of cyclic homology, and points out some interesting consequences. In the second part, the author proves a sort of localization theorem for cyclic homology: ``under suitable flatness hypotheses an exact sequence of derived categories yields a long exact sequence in cyclic homology''. In order to prove this, the author considers an interesting definition of cyclic homology of ``model categories''. Model categories are exact DG categories satisfying certain conditions imposing nice properties on the homomorphism spaces. This gives better flexibility than what one gets if one works with categories of DG modules only. For instance, the author shows that sequences of model categories yielding short exact sequences of stable categories can (for the present purposes) be assumed to be exact. It is explained how the localization theorem is useful when considering localizations of rings with respect to multiplicative subsets admitting a calculus of fractions, and also how analytic isomorphisms yield isomorphisms in relative cyclic homology. The casual reader should note that much material from the last section is used throughout the paper, and it makes sense to take a quick glance at it to get the relevant notation and terminology used in the paper.
scientific article; zbMATH DE number 1121668
scientific article