This is more a remark then an answer.
One important application of projective/locally free resolution is the computation of intersection numbers. Let $X$ be a smooth projective variety over a field $k$, and $Z_1,Z_2$ subschemes of $X$ of complementary dimension. Let $L^*$ be a locally free resolution of $\mathcal{O}_{Z_1}$ then the intersection number $(Z_1 . Z_2)$ equals $\chi(X,L^* \otimes \mathcal{O}_{Z_2})$ (Serre's formula).
While the geometric meaning of $(Z_1 . Z_2)$ is clear (at least over the complex numbers): take a generic topological perturbation and count transversal intersections with orientation; the interpretation of $\chi(X,L^* \otimes \mathcal{O}_{Z_2})$ is less so.
We somehow trade the torsion sheaf $\mathcal{O}_{Z_1}$ for the more "global" object $L^*$ (with $Supp(L^i)=X$). We certainly need some global information about $X$ to compute the intersection, as the generic perturbations take place in $X$.
Note that if $Z_1 \sim Z_1'$ is an algebraic generic perturbation, such that $Z_1' \cap Z_2$ is transversal (and zero-dimensional), then $\chi(Z_1' \otimes Z_2)=h^0(Z_1' \otimes Z_2)=(Z_1 . Z_2)$. So there is a formal analogy between $L^*$ and a generic perturbation of $Z_1$.