Homological dimensions of smooth crossed products

In this paper we provide upper estimates for the global projective dimensions of smooth crossed products \(\mathscr{S}(G, A; \alpha)\) for \(G = \mathbb{R}\) and \(G = \mathbb{T}\) and a self-induced Fréchet-Arens-Michael algebra \(A\). In order to do this, we provide a powerful generalization of methods which are used in the works of Ogneva and Helemskii.