Homological dimensions and Van den Bergh isomorphisms for nuclear Fréchet algebras

摘要

We prove the equation w.dgA = w.dbA for every nuclear Fréchet-Arens- Michael algebra A of finite weak bidimension, where w.dgA is the weak global dimension and w.dbA the weak bidimension of A. Assuming that A has a projective bimodule resolution of finite type, we establish the estimate dbA 6 dgA + 1, where dgA is the global dimension and dbA the bidimension of A. We also prove that dgA = dbA = w.dgA = w.dbA = n for all nuclear Fréchet-Arens-Michael algebras satisfying the Van den Bergh conditions VdB(n). As an application, we calculate the homological dimensions of smooth and complex-analytic quantum tori.