Abstract:
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.
Keywords:nuclear Fréchet algebra, global dimension, bidimension, Van den Bergh isomorphisms, Hochschild homology.
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