Suppose that $X = \mathbb{P}^n_k$ and $G$ is a coherent sheaf on $X$.

**Question:** Is there a way to determine *some* integer $n_0$ such that $H^1(X, G \otimes O_X(n)) = 0$ for all $n \geq n_0$? (obviously $n_0$ depends on $G$)

Of course, the higher cohomologies would be very interesting as well, but perhaps the above is simpler. One can also ask this for the relative case as well $f : Y \to X$ a projective morphism.

What I'd be particularly interested in is something that can be implemented into a computer.

In my particular case, I know I only have to compute some other things until Serre vanishing hits, at that point, I know I can stop computing. Unfortunately, I don't know how to tell when I've *arrived*. You can assume I have a presentation of my module $G$ if it helps.

**PS:** *Actually, what I really want is vanishing of* $$R^1 f_* (G(-nE))$$ *for a blow-up $f : Y \to X$ with $E$ the pullback of the blownup ideal.*