# Twisting sheaf of projective space

Let $$A$$ be a ring, $$S=A[x_{0},...,x_{n}]$$, and $$X=$$Proj $$(S)=\mathbb{P}_{A}^{n}.$$ Hartshorne defines the twisting sheaf $$\mathcal{O}_{X}(n)=S(n)^{\thicksim}$$. Since $$\mathcal{O}_{X}(n)|_{D+(x_{i})}$$ is isomorphic to $$S(n)_{(f)}^{\thicksim}$$ on Spec $$S_{(f)}$$ and for any $$n$$, $$S(n)_{f}$$ is isomorphic to $$S_{(f)}$$ as $$S_{(f)}-$$module, it look likes $$\mathcal{O}_{X}(n)$$ is isomorphic to $$\mathcal{O}_{X}(m)$$ as sheaf of modules. However, I think they are definitely not isomorphic. Where I make a mistake? And how to proof if $$n\not =m$$, $$\mathcal{O}_{X}(n)\not \backsimeq \mathcal{O}_{X}(m)$$. In general, how to know two locally free sheaf of the same rank are not isomorphic?

• Take their global sections. – Youngsu Apr 2 at 17:38
• @Youngsu Could you tell me how to compute their global section? I can not see that their global sections are not isomorphic – Mike Apr 3 at 1:08
• This is done in Hartshorne's book of Algebraic geometry. Look at Proposition II.5.13. – Youngsu Apr 3 at 15:08