A sheaf-theoretic version of the proj construction?

Question

Recall that $\operatorname{Sch}$ can be identified with the subcategory of (Zariski-)locally representable (by an affine) étale sheaves on $\operatorname{CRing^{op}}$. In this case, $\operatorname{Spec}(-):\operatorname{CRing^{op}}\to \operatorname{Sch}$ is simply the co-Yoneda embedding $R\mapsto \operatorname{Hom}(R,-)$ (which makes sense since the étale topology is subcanonical and $\operatorname{Hom}(R,-)$ is obviously locally affine).

Is there a nice "sheaf-theoretic" description of $\operatorname{Proj}:\operatorname{(Gr_{{\mathbf Z}_{\geq 0}}CRing)^{op}}\to \operatorname{Sch}$ (I've only seen $\operatorname{Proj}$ for nonnegative integral grading. If we can use more exotic gradings, I guess I'd be interested in that too). Hoping for it to be as nice as $\operatorname{Spec}$ seems like a bit of a pipe dream, but I'm wondering if there is a nicer way to describe it than Hartshorne's approach, which feels rather arbitrary.

Edit: To clarify, I'm looking for a construction $\operatorname{Proj}:\operatorname{(Gr_{{\mathbf Z}_{\geq 0}}CRing)^{op}}\to \operatorname{Sh}_{\acute{et}}(\operatorname{CRing^{op}})$, which we can then see lands in $\operatorname{Sch}$ by showing that we can cover it with Zariski-open affines.

Answer 1

Perhaps I've not understanding your question, but it sounds like you're asking "What is the functor of points of a Proj?"

The answer, of course, is the functor that sends a scheme $X$ to the set of line bundles $L$ on that scheme equipped with a graded map of $R$ to $\Gamma(X;\oplus_{n\geq 0} L^n)$ whose image generates $L^n$ as a $\mathcal{O}_X$-module. The affine open sets are given by the subset where a positive degree element $r$ of $R$ is sent to non-vanishing section; these open sets are easily seen to be the Spec of the degree 0 part of $R_r$, the localization of $R$ at $r$.

EDIT: Let me incorporate some of the things I said below: if you want to just work with rings, then replace "line bundle" with "invertible projective" and $\Gamma(X;\oplus_{n\geq 0} L^n)$ with "tensor algebra over A."

If you're OK with sheafifying, you can do something simpler, which is assume that your line bundle is trivial (but not with a fixed trivialization), i.e. that your invertible projective is free of rank 1 (but not canonically isomorphic to A).

A graded map from $R$ to the tensor algebra of a rank 1 free module is the same as a map from $R\to A$, after you pick an isomorphism of that module to $A$. However, you have to identify maps that come from the same map to the tensor algebra under different isomorphisms (i.e. mod out by the action of the units of A), and throw out maps where the images of the degree 1 elements don't generate A (this is why you get a Proj-ish thing).

Answer 2

I've learned a functorial description of the Proj-construction in a script on algebraic geometry by Marc Nieper-Wißkirchen. Unfortunately I cannot find it online anymore. The script is heavily influenced by the book Groupes algébriques by Demazure and Gabriel, thus you will probably also find it here.

In the following, functors are always functors from (Ring) to (Set). Affine schemes are precisely the representable ones, and general schemes are obtained by gluing these appropriately. Consider the group functor $G_m$, which is given by $R \mapsto R^*$. Observe that an affine scheme together with an action of $G_m$ is the same as a graded ring. Here, the grading is indexed by integers. If $X$ is a scheme together with a group action, then there is a largest invariant open subscheme $X^+$ , such that the stabilizers are finite (the latter being defined pointwise). In the above example $X - X^+$ is cut out by the irrelevant ideal of the grading.

Now we have the following theorem:

Let $G_m$ act on an affine scheme $X$. Then the categorical quotient $Proj(X) := X^+ / G$ exists.

The proof can be stated in a pure functorial way, but of course basically you just write down the usual Proj construction for a graded ring. Nevertheless, it is very enlightening. The whole basic theory about Proj can be developed in this functorial setting (field-valued points, projective space, quasi-coherent modules, Serre twists).

Answer 3

Try EGA chapter II section 3, especially 3.7 if you want to know the functor of points of proj.

Answer 4

In one special case of the Proj construction the functor is easy to write down, namely in the case of $\mathbb{P}^n$. The functor you get in $Sh(Cring^{op})$ is:

$R$ gets mapped to split inclusions of $R$-modules $N \rightarrow R^{n+1}$ where $N$ is locally free of rank 1.

A Zariski open of this functor is defined by taking the subfunctor

Inclusions $j: N \rightarrow R^{n+1}$ such that composition with $p_i R^{n+1} \rightarrow R$ is an isomorphism. This gives you your n+1 Zariski opens.

I think the general case should be something similar.

Answer 5

Do you know Stacks Project, section "27 (=Tag 01LE) Constructions of Schemes"? In it you find Proj of a general (sheaf of) graded ring(s), its functor of points, quasi-coherent modules...