Have I made an error in my reasoning?
If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} }(N_{\mathfrak{m}} ,M_{\mathfrak{m}} )\cong Ext_A(N,M)\otimes_A A_{\mathfrak{m}} $ for every $N,M \in _AMod$ then:
$$D(A)\geq D(A_{\mathfrak{m}} )=\dim(A),$$
where $D$ is the global dimension of $A$ and $\dim(A)$ is the Krull dimension of $A$.
Pick an $A_m$-module $M$, view it as an $A$-module and find a projective resolution. Tensor it with $A_m$: the resulting complex is an $A_m$-projective resolution of $M$. It follows immediately that the global dimension of $A$ bounds that of its localizations.