Noncommutative Fibrations

We show that faithfully flat smooth extensions are reduced flat, and therefore, fit into the Jacobi-Zariski exact sequence in Hochschild homology and cyclic (co)homology even when the algebras are noncommutative or infinite dimensional. We observe that such extensions correspond to 'etale maps of affine schemes, and we propose a definition for generic noncommutative fibrations using distributive laws and homological properties of the induction and restriction functors. Then we show that Galois fibrations do produce the right exact sequence in homology. We then demonstrate the versatility of our model on a geometro-combinatorial example. For a connected unramified covering of a connected graph $G^{\mathrm{\prime }}\to G$, we construct a smooth Galois fibration ${\mathcal{A}}_G\subseteq {\mathcal{A}}_{G^{\mathrm{\prime }}}$ and calculate the homology of the corresponding local coefficient system.