Inertial Hopf-cyclic homology

We construct a functor from the category of comodule algebras over a Hopf algebra to the category of stable anti-Yetter-Drinfeld modules over that Hopf algebra. In the commutative case it reproduces the inertia groupoid for a given group action groupoid. Given any right comodule algebra and any right module coalgebra with a grouplike element, the resulting Hopf-cyclic object with coefficients in that stable anti-Yetter-Drinfeld module can be conceptually understood as a Hopf-cyclic quantization of the cyclic nerve of the group-action groupoid. We call it the inertial Hopf-cyclic object. Next, we construct a natural map from the Kadison cyclic object of a given comodule algebra regarded as an extension of its subalgebra of invariants of the coalgebra coaction to the inertial Hopf-cyclic object. This is a quantization of the characteristic map from the cyclic nerve of the group-action groupoid to the cyclic Alexander-Spanier space. We show that if that coalgebra coaction satisfies the Galois condition, the natural map of cyclic objects becomes an isomorphism. This means the invertibility of the quantized characteristic map for principal bundles. In the Hopf-Galois case we find a non-trivial identification of our map with a well known isomorphism of Jara-Stefan.