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?

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

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