# Construction of the global $\mathbf{Proj}$

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!

• Proj is a very abstract, general and subtle construction. Probably the best introductory source is Eisenbud and Harris' The Geometry of Schemes. They have a good discussion of Proj and its relation to blowing-up. Commented Jan 16, 2012 at 17:02
• I don't remember the examples given in The Geometry of Schemes, but you should try working out two examples by hand without looking them up. What happens if $X=Spec k$ for some field? Second, for an actual blow-up, you should be able to explicitly write down the ideal sheaf of a point on $\mathbb{P}^1$, so try following the definitions through for blowing-up a point.
– Matt
Commented Jan 17, 2012 at 17:50
• The global Proj is not so abstract, but given your questions about it, it sounds as if you are not sufficiently far along in your study of quasicoherent sheaves and related topics to follow the construction. Before thinking about global Proj, have you worked through the exercise in Hartshorne (somewhere in the first half of Chapter II) about global Spec. The goal of the exercise is to establish an equivanlence between affine morphism $Y \to X$ and quasi-coherent sheaves of $\mathcal O_X$-algebras on $X$. If you haven't worked through this exercise you should. (Just as it wouldn't make ... Commented Feb 17, 2012 at 2:33
• ... sense to study Proj of a graded ring before understanding the concept of Spec of a ring). Regards, Commented Feb 17, 2012 at 2:33

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 $\mathrm{Proj}\mathcal S(U_f)$.

• $A'$ should be an $A$-agebra. Commented Apr 2, 2020 at 14:00

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.