Homological dimensions and Van den Bergh isomorphisms for nuclear Fréchet algebras

We prove the equation $\operatorname{w{.}dg} A=\operatorname{w{.}db} A$ for every nuclear Fréchet–Arens–Michael algebra $A$ of finite weak bidimension, where $\operatorname{w{.}dg} A$ is the weak global dimension and $\operatorname{w{.}db} A$ the weak bidimension of $A$. Assuming that $A$ has a projective bimodule resolution of finite type, we establish the estimate $\operatorname{db}A\le\operatorname{dg}A+1$, where $\operatorname{dg} A$ is the global dimension and $\operatorname{db} A$ the bidimension of $A$. We also prove that $\operatorname{dg}A=\operatorname{db}A=\operatorname{w{.}dg}A= \operatorname{w{.}db} A=n$ for all nuclear Fréchet–Arens–Michael algebras satisfying the Van den Bergh conditions $\operatorname{VdB}(n)$. As an application, we calculate the homological dimensions of smooth and complex-analytic quantum tori.