INERTIAL HOPF-CYCLIC HOMOLOGY

We construct, study and apply a characteristic map from the relative periodic cyclic homology of the quotient map for agroup action to the periodic Hopf-cyclic homology with coefficients associated with the inertia of the action. This characteristic map comes from its noncommutative-geometric, or quantized, counterpart. A crucial ingredient herein is the construction of appropriate quantization of the cyclic nerve of the action groupoid. It is the cyclic object related to inertia as the Connes-cyclic dual of a Hopf-cyclic object with coefficients in some stable anti-Yetter–Drinfeld module quantizing the Brylinski space. For the Hopf-Galois quantization of the case of trivial inertia, we find a non-trivial identification of our characteristic map with the well known isomorphism of Jara and Stefan. In the presence of non-trivial inertia, for an analytic ramified Galois double cover, we show that the cokernel of our characteristic map, which can be interpreted as an invariant of inertia modulo the topology of the maximal free action, is supported on the branch locus of the quotient map. Finally, we use our inertial Hopf-cyclic object to construct a new invariant of finite-dimensional algebras. (Based on joint work with Serkan Sutlu.)