Homological dimensions of analytic Ore extensions

TL;DR

This paper investigates the homological dimensions of analytic Ore extensions, extending classical algebraic results to the setting of holomorphic and smooth crossed products of topological algebras.

Contribution

It generalizes the bounds on global dimensions from algebraic Ore extensions to analytic and topological contexts, including holomorphic and smooth crossed products.

Findings

Global dimension bounds extend to holomorphic Ore extensions.

Results apply to smooth crossed products by Z.

Provides new insights into homological properties of topological algebras.

Abstract

If $A$ is an algebra with finite right global dimension, then for any automorphism $\alpha$ and $\alpha$-derivation $\delta$ the right global dimension of $A[t; \alpha, \delta]$ satisfies \[ \text{rgld} \, A \le \text{rgld} \, A[t; \alpha, \delta] \le \text{rgld} \, A + 1. \] We extend this result to the case of holomorphic Ore extensions and smooth crossed products by $\mathbb{Z}$ of $\hat{\otimes}$-algebras.