Can someone explain the construction of the global $\mathbf{Proj}$ to me? Although this question has been asked here, I still have several questions.
- For each open affine subset $U = \mathrm{Spec} A$ of $X$, let $\mathcal{S}(U) = \Gamma(U, \mathcal{S}|_{U})$, which is a graded $A$-algebra. Then we have a natural morphism $\pi_{U}: \mathrm{Proj} \mathcal{S}(U) \rightarrow U$.
This map is from $A \rightarrow \mathrm{Proj} \mathcal{S}(U)_{f}: a \rightarrow a/1 $, right?
- Let $f \in A$ and $U_{f} = \mathrm{Spec} A_{f}$. Since $\mathcal{S}$ is quasi-coherent we have $\mathrm{Proj} \mathcal{S}(U_{f}) \cong \pi_{U}^{-1}(U_{f})$.
I don't really understand the explanation under that question. Why $\mathcal S(U_f)=\mathcal S(U)_f$? And why we need $\mathcal{S}$ to be quasi-coherent?
- How to glue the invertible sheaves $\mathcal{O}(1)$ on each $\mathrm{Proj} \mathcal{S}(U)$ to get an invertible sheaf $\mathcal{O}(1)$ on $\mathbf{Proj}S$
Can someone explain more explicitly? Thank you very much!
Here is my understanding:
Suppose $\mathcal{S}|_{U}=\tilde{M}$, then $\mathcal{S}|_{U}(U)_f=M_f=\mathcal{S}|_{U}(U_f)$. Then we have this commutative diagram: $\require{AMScd}$ $$\begin{CD} \mathrm{Proj}\mathcal{S}(U_f)=\mathrm{Proj}\mathcal{S}(U)\times_{\mathrm{Spec}A}\mathrm{Spec}\,\mathcal A_f @>>> \mathrm{Spec}\,\mathcal A_f\\ @VVV @VVV \\ \mathrm{Proj}\mathcal{S}(U) @>>> \mathrm{Spec}\,\mathcal A. \end{CD}$$