I've been having a hard time with graded objects in algebraic geometry for some time. Lately I realized a lot of my difficulties come from not having any idea at all of where graded objects live. Having no idea what category they come from I am forced to swallow a lot of ad hoc constructions which are hard to internelize. So I decided to attack these core issues.
Alas, I can't seem to find a source which treats the basics of categories of graded modules/algebras. I didn't find even the basic definition for a category in either cases.
Here's what I have so far:
- Rings can be viewed as monoids in $\mathsf{Ab}$, modules are modules over monoids and algebras are monoids in the category of modules.
This statement generalizes imediately to the category of graded abelian groups. The jist here (from what i've been able to gather) is we have (at least?) two choices for the braiding in the symmetric monoidal category of graded abelian groups (kozul sign rule).
I'm still having trouble though with the down to earth understanding. Here are couple of questions:
- How well behaved is the subcategory of positively/non-negatively graded modules/algebras?
- What is the graded tensor product? (pushout of graded algebras).
- Is the category of graded modules over a graded algebra abelian?
- Is graded localization of modules/rings/ algebras universal in some sense?
- Is the $\mathsf{Proj}$ functor adjoint to something? If so what equivalence of subcategories does it induce?
All these would be much easier to answer once I have a clear definition of those categories and some basic properties.
There is a symmetric monoidal abelian category of $\mathbb{Z}$-graded abelian groups, where the monoidal structure ("graded tensor product") is given by convolution
$$(V \otimes W)_n \cong \bigoplus_{i+j=n} V_i \otimes W_j$$
and the symmetry is the obvious thing (not the Koszul thing, that's completely irrelevant to Proj). This is a special case of Day convolution.
A graded ring is a monoid, and a graded (commutative ring) is a commutative monoid (note that when most people say "graded commutative ring," say in algebraic topology, they want the Koszul symmetry, not the obvious one). The graded tensor product of graded rings is just the graded tensor product of graded abelian groups with a natural ring structure. (In particular if the rings aren't commutative then this is not the coproduct.) Graded modules over a graded ring form an abelian category as expected. I haven't really thought about graded localization.
This is mostly irrelevant to Proj, which you should be thinking of much more geometrically. Geometrically the idea is the following. Suppose $X \subseteq \mathbb{P}^n$ is a closed subvariety of projective space. Then we can take its preimage in $\mathbb{A}^{n+1} \setminus \{ 0 \}$ under the projection
$$\mathbb{A}^{n+1} \setminus \{ 0 \} \to \mathbb{P}^n$$
given by quotienting by the action of the multiplicative group $\mathbb{G}_m$. After reinserting the origin, the result is a closed subvariety of $\mathbb{A}^{n+1}$ called the affine cone of $X$, which I'll denote $\widetilde{X}$. The $\mathbb{G}_m$ action on $\mathbb{A}^{n+1}$ extends to $\widetilde{X}$, and to get $X$ from $\widetilde{X}$, remove the origin and then quotient by the $\mathbb{G}_m$ action.
How can we say this algebraically? The ring of functions on $\widetilde{X}$ is the homogeneous coordinate ring of $X$ with respect to the embedding above (which really depends on this choice of embedding; it is not intrinsic to $X$). The $\mathbb{G}_m$ action manifests itself in the grading on the homogeneous coordinate ring, taking invariants corresponds to looking at homogeneous prime ideals, and removing the origin corresponds to ignoring the "irrelevant" ideal.
So Proj is a much less general and clean construction than it seems at first glance. It really has a relatively modest goal, which is to give an algebraic way to write down and work with projective varieties. Among other things, as mentioned in the comments, this business with removing the origin prevents it from being functorial with respect to arbitrary maps of graded rings. This is really a special case of the theory of GIT quotients.