Proj for rings graded by different things then $\mathbb N$ ?

Question

Given a commutative, $\mathbb N$-graded ring, one can associate to it a scheme via the $Proj$ construction.

What happens if one tries to copy this procedure but instead of $\mathbb N$ with another indexing gadget (say commutative monoid) ?

Some thoughts about this: Considering projective varieties is roughly the same as studying affine varieties equivariant under the multiplicative group. So I would guess that replacing $\mathbb N$ by something else corresponds to replacing the multiplicative group by something else.

Answer 1

Weighted projective spaces $\mathbb{P}(a_1,\ldots,a_n)$ are examples where a grading other than the standard grading is used. In general you can study gradings coming from any finitely generated abelian group, and this grading gives rise to a torus action on the ring. The Proj you speak of is then a GIT-quotient of $Spec R$ by this torus action (if you are familiar with GIT, the choice of linearization of the action corresponds to a choice of irrelevant ideal). This GIT-quotient construction is completely analogous to the usual construction $$ \mbox{Proj} k[x_0,\ldots,x_n]=\left(\mbox{Spec} k[x_0,\ldots,x_n]-V(x_0,\ldots,x_n)\right) // \mathbb{G}_m $$ The best reference I can give for this stuff is Chapter 1 of the book "Cox rings" by Arzhantsev, Derenthal, Hausen and Antonio Laface. Other nice references are

The Homogeneous Coordinate Ring of a Toric Variety by Cox, which deals with Toric varieties

Lectures on invariant theory by Dolgachev, which is a nice introduction to quotients in algebraic geometry.

Answer 2

First let's revisit the usual case. The $\mathbb N$-grading on $R$ says that $Spec\ R$ is a cone, and there is a map $R \to R_0$. We can rip out $Spec\ R_0$ from $Spec\ R$, take the quotient, and get a reasonable space $Proj\ R$.

If we grade instead by say $({\mathbb N})^k$, then there are $k$ analogous quotients of $R$, and $k$ closed subschemes to rip out before we divide by $({\mathbb G}_m)^k$. This comes up if e.g. $R = Fun(G/N)$, with the torus acting on the right (through the maximal unipotent group $N$), and the corresponding quotient is $G/B$.

Some of the things I would look up here are "Cox coordinate ring of a toric variety" and "Mori dream space", but I don't know the references well enough to suggest a best place.