Let $X$ be a smooth projective variety over algebraically closed field of characteristic zero. In one of the versions of definition of tilting object $\mathcal{F}$ on $X$ there is a requirement that global dimension of endomorphism algebra $\operatorname{End}_X(\mathcal{F})$ is finite.
I'm interested in more general situation, let $\mathcal{F}$ be some coherent sheaf on $X$. I would like to know what are methods for calculating global dimension of $\operatorname{End}_X(\mathcal{F})$?