Hi Mathoverflow. This question is about building intuition for the Proj construction. When I first started learning about schemes, I found the construction of the structure sheaves on Spec and Proj very confusing. However, after enough time had passed I began to understand the construction for Spec: It is sort of an algebraic partition of unity argument. The construction of the structure sheaf on Proj is still very mysterious to me. For completeness it goes something like this:

Let $G$ be a graded ring. Take $ f \in G $ homogeneous of degree $ d \geq 1 $. We have a homeomorphism $D_+(f) \cong{\rm spec} \, G_{(f)} $ which sends $D_+(fg)$ to $D(g^{\deg f} / f^{\deg g})$. It first expands the prime to $G_f$ and then contracts the result to $G_{(f)}$. Motivated by this, we can define $$ \mathscr{O}_{{\rm Proj}G}(D_+(f)) = G_{(f)}$$ and prove that the map $G_{(f)} \to G_{(fg)}$ defined by $ a / f^n \mapsto a g^n / (fg)^n$ is localization at $ g^{\deg f} / f^{\deg g}$.

I can fill in the details but they are messy and I have very little idea what the details actually mean. Thinking about $ \mathbb{P}^n $ as a variety, I understand why $D_+(X_0)$ should be $ {\rm spec} \mathbb{C} [X_1/X_0,X_2/X_0, \dots, X_n / X_0] $ but I don't really understand how this intuition translates into the commutative algebra which is boxed above.

Also, for homogeneous elements of degree higher than $1$ I have no idea what is going on. I understand that geometrically the veronese map should be involved but I don't understand how that intuition translates into the messy proof which I am able to write down.

Question: Is anyone able to explain the idea behind this construction? Note that this explanation could lie in either the realm of algebraic geometry or commutative algebra.

It is very possible that there is not a nice way to think about this construction which would make me sad because it is so fundamental. I hope that there are some good responses which help me fix my ignorance!

EDIT: I just went through the proof again and wrote up the details.

Let $ G$ be a graded ring. The first order of business is to explain why we have a homeomorphism $ D_+(f) \cong {\rm Spec} G_{(f)} $ when $ f \in G $ is homogeneous with degree $d \geq 1 $. Fix $ \mathfrak{p} \in D_+(f)$. Then $ \mathfrak{p}$ is the generic point of some irreducible closed subset $Z$ in $ {\rm Spec} G$. Since $ f \not\in \mathfrak{p}$ it follows that $ G_f \mathfrak{p}$ is the generic point of the irreducible closed subset $ Z \cap {\rm Spec} G_f$ in $ {\rm spec} G_f$. Assume that $g \in G $ is homogeneous. Then $ g $ vanishes at $ \mathfrak{p} $ iff $ g / 1 $ vanishes at $ G_f \mathfrak{p}$ iff $g^{\deg f} / f^{\deg g}$ vanishes at $ G_f \mathfrak{p}$ iff $g^{\deg f} / f^{\deg g} \in G_f \mathfrak{p} \cap G_{(f)} $. This proves that the map $D_+(f) \to {\rm Spec} G_{(f)}$ is injective. We need to prove that the map is surjective. Let $ \mathfrak{q}$ be a prime ideal in $G_{(f)}$. Motivated by the above argument, we define $$ \mathfrak{p}_n = \{ g \in G_n : g^{\deg f} / f^{\deg g} \in \mathfrak{q} \} $$ Then $ \mathfrak{p} = \oplus_{n \geq 0} \mathfrak{p}_n \in D_+(f)$ maps to $ \mathfrak{q} $. Since $ g $ vanishes at $ \mathfrak{p} $ iff $g^{\deg f} / f^{\deg g} \in G_f \mathfrak{p} \cap G_{(f)} $ the map is a homeomorphism which sends $D_+(fg)$ to $D(g^{\deg f} / f^{\deg g})$. All that remains is to prove that the map $ G_{(f)} \to G_{(fg)}$ defined by $ a / f^n \mapsto a g^n / (fg)^n$ is the localization map. From the affine case we know that $ G_{f} \to G_{fg}$ is the localization map at $ g / 1 $, therefore the only non units which $ G_{(f)} \to G_{(fg)}$ sends to units are powers of $ g^{\deg f} / f^{\deg g}$. I suspect that this last bit can be made rigorous.

quotient$R \to R_0$. This only works because $R$ is $\mathbb N$-graded, not just $\mathbb Z$-graded. $\endgroup$