  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 Fr\'echet-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.
