This is from Mumford-Oda's Algebraic Geometry 2, And here is pdf of chapter 7-8. https://www2.math.upenn.edu/~chai/624_08/mumford-oda_chap7-8.pdf
My question is on page 243, I can't see why $H^{i+1}(\mathcal{K}_1)$ could be computed as shown in the text. I try to use long exact sequence of $$0 \to \mathcal{F}_1 \to \mathcal{F}\to \mathcal{F}/\mathcal{K}_1 \to 0$$, but the only acyclic sheaf is $\mathcal{F}(d)$, which is not in this short exact sequence...
Thanks for any help.