| Abstrakt: |
We prove the equation $ \operatorname{w{.}dg} A=\operatorname{w{.}db} A$ for every nuclear Frechet-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 Frechet-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. |
