The global dimension of a ring $R$ is the supremum of the projective dimensions of it's $R$-modules.
$$\dim (R)=\sup \{\dim_\mathrm{proj}(M):M \in R\text{-mod} \}$$
I'd like to have some geometric picture for the meaning of this number.
If $R$ is noetherian we have that "projective", "locally-free", and "flat" are all equivalent. Any module over $R$ can be considered as a sheaf of$\mathcal{O}_S$-modules where $S=\operatorname{Spec} R$.
A projective resolution of a module $M$ could be thought of as fitting $M$ into the end of an exact complex of vector bundles (locally-free sheaves) over $S$.
This suggests the following story for arbitrary (paracompact hausdorff) topological spaces.
Definition 1: An $(r,k)$-resolution for a vector bundle morphism $f: E_0 \to F_0$ is an exact complex of vector bundles:
$$0 \to E_r \to \cdots \to E_{2} \to E_{1} \to E_0 \to F_0 \to F_1 \to F_2 \to \cdots \to F_k \to 0$$
Definition 2: The left dimension a morphism $f: E_0 \to F_0$ is the infimum over all lengths of resolutions:
$$\dim_{\leftarrow} (f) = \inf \{n : f \text{ has an $(n,k)$-reolution for some $k\in \mathbb{N}\cup \{\infty\}$} \}$$ Dualizing we get the right dimension $\dim_\rightarrow(-)$.
Mimicking we get "homological dimensions" for $X$ by taking the supremum:
Definition 3: The left "global" dimension of a space $X$ (or a smooth manifold when taking the morphisms in the suitable category) is given by: $$\dim_{\leftarrow}(X) = \sup \{\dim_{\leftarrow} (f) : f \in \mathsf{Mor}(\operatorname{Vect}(X))\}$$ Dualizing we get the right "global" dimension $\dim_\rightarrow(X)$.
(1) Is the "global" dimension in definition (3) a known/useful concept? (If not for topological spaces maybe for smooth manifolds?)
(2) What are some (more direct perhaps) geometric interpretations for the global dimension of a ring?