Noncommutative Fibrations

TL;DR

This paper explores noncommutative fibrations, establishing their homological properties, defining a new class of fibrations, and demonstrating their application through a graph covering example.

Contribution

It introduces a definition for noncommutative fibrations using homological properties and distributive laws, extending classical concepts to noncommutative and infinite-dimensional algebras.

Findings

Faithfully flat smooth extensions are reduced flat and fit into exact sequences in Hochschild and cyclic homology.

Galois fibrations produce the correct homological exact sequences.

Application to a graph covering example demonstrates the model's versatility.

Abstract

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 $G'\to G$, we construct a smooth Galois fibration $\mathcal{A}_{G}\subseteq\mathcal{A}_{G'}$ and calculate the homology of the corresponding local coefficient system.