Hartshorne book (III.5.1) say that the natural map $$H^0(\mathbb{P}^r_{A},\mathcal{O}(n)) \times H^r(\mathbb{P}^r_{A},\mathcal{O}(-n-r-1)) \rightarrow H^r(\mathbb{P}^r_{A}, \mathcal{O}(-r-1))$$ is a perfect pairing of finitely generated free $A$-modules for each $n\in \mathbb{Z}$
I heard that we can prove $H^d(\mathbb{P}^r_{A}, \mathcal{O}(n))=0$ for $n\geq -d$ using the above fact... I try this proof... but I can't prove..
