Let $X$ be a smooth variety, $x$ a closed point, and $\mathcal{O}_x$ the skyscraper sheaf of the residue field $k(x)$ at $x$. Recall that the homological dimension of a complex of coherent sheaves $\mathcal{F}^\bullet$ is the smallest $n$ such that there exists a quasi-isomorphism between $\mathcal{F}^\bullet$ and a complex of locally free sheaves of length $n+1$. (A quasi-isomorphism of complexes is a morphism such that the induced homology maps are isomorphisms.)
How can one show that the homological dimension of $\mathcal{O}_x$ (considered as a complex of length one) is equal to the dimension $n$ of $X$?