Homological dimensions of smooth and complex analytic quantum tori

TL;DR

This paper surveys homological dimensions of various quantum tori, revealing that smooth and complex analytic versions have global dimension equal to their dimension, contrasting with algebraic cases.

Contribution

It establishes that smooth and complex analytic quantum tori have global dimension equal to their dimension, providing new insights into their homological properties.

Findings

Smooth and complex analytic quantum n-tori have global dimension n.

Algebraic quantum n-tori have global dimension 1 in the generic case.

Formulation of general theorems on homological dimensions of nuclear Fréchet algebras.

Abstract

We survey some results on homological dimensions of the algebraic, complex analytic, and smooth quantum tori. Our main theorem states, in particular, that the smooth and the complex analytic quantum n-tori have global dimension n. This contrasts with the result of McConnell and Pettit (1988) who proved that, in the generic case, the algebraic quantum n-torus has global dimension 1. In this connection we also formulate some general theorems on homological dimensions of nuclear Fr\'echet algebras.