Construction of the global $\mathbf{Proj}$

Question

I have many questions about the very abstract concept of global $\mathbf{Proj}$. I am following Hartshorne's book Algebraic Geometry, where this concept is on II.7, page 160.

Let $(X, \mathcal{O}_{X})$ be a noetherian scheme and $\mathcal{S} = \bigoplus_{d \geq 0} \mathcal{S_{d}}$ a quasi-coherent sheaf of $\mathcal{O}_{X}$-modules, which has a structure of a sheaf of graded $\mathcal{O}_{X}$-algebras. Assume that $\mathcal{S}_{0} = \mathcal{O}_{X}$, that $\mathcal{S}_{1}$ is a coherent $\mathcal{O}_{X}$-module and that $\mathcal{S}$ is locally generated by $\mathcal{S}_{1}$ as an $\mathcal{O}_{X}$-algebra. Why this implies $\mathcal{S}_{d}$ is coherent for all $d \geq 0$?

For the construction of the global $\mathbf{Proj}$, for each 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 $\pi_{U}: \mathrm{Proj} \mathcal{S}(U) \rightarrow U$. Let $f \in A$ and $U_{f} = \mathrm{Spec} A_{f}$. Hartshorne says that since $\mathcal{S}$ is quasi-coherent we have $\mathrm{Proj} \mathcal{S}(U_{f}) \cong \pi_{U}^{-1}(U_{f})$. Why? I can not see that.

In my opinion, the global $\mathbf{Proj}$ is a very abstract concept that I can not get any concrete example of blowing up. Do you know a book where I can find concrete examples?

Thank you!

Answer 1

First if $\mathcal S$ is locally generated by $\mathcal S_1$ (as algebra), by definition $\mathcal S_d$ is generated by the product of $d$ elements of $\mathcal S_1$ ($\mathcal S$ is a quotient of the symetric algebra $\mathrm{Sym}(\mathcal S_1)$ and it is enough to think locally). Then it is clear that $\mathcal S_d$ is locally finitely generated, hence coherent because $X$ is noetherian.

For the second part of your question, notice that $\mathcal S(U_f)=\mathcal S(U)_f$, and that Proj commutes with tensor product. More precisely, if $B$ is a homogeneous algebra over a ring $A$ and $A'$ is an $R$-algebra, we can endowed $B\otimes_A A'$ with the natural graduation coming from that of $B$ (trivial graduation on $A'$. Then one can show that $$\mathrm{Proj}(B\otimes_A A')=\mathrm{Proj}(B)\times_{\mathrm{Spec}A}\mathrm{Spec} A'.$$ Now taking $A'=A_f$ and $B=\mathcal S(U)$ will give you the isomorphe on $\mathrm{Proj}\mathcal S(U_f)$.

Answer 2

This answer is geared towards people who have gotten used to Proj after 12 years, but want a cleaner perspective.

Relative Spec is a more common and intuitive tool. Usually, one does relative spec (RS from now) for a sheaf $F$ on a scheme $X$.

However, one can also take a stack $X$ and use RS to obtain a stack $Y \to X$. Taking $X = BG_m$, a sheaf on $BG_m$ is a graded abelian group, and a commutative algebra object there is a graded ring.

Thus applying $RS$ we obtain a stack $Y$ with a map $Y \to BG_m$, and of course we can recover $Y$ via taking global sections of the line bundle.

How does this lineup with the fact that $Proj$ can give the same result sometimes? Because $Proj$ is the coarse space of the STACK.