Ravi Vakil's notation $\operatorname{Proj}_A S$

Question

I saw the notation $\operatorname{Proj}_A S$ for a ring $A$ and a graded $\mathbb{Z}^{\geq 0}$-graded ring $S$ being used in Ravi Vakil's notes on Algebraic Geometry. I first came across the notation at the start of $\S 17.2$. Since $S$ does not have any underlying $A$-algebra structure, I do not know what this subscript notation actually means. I was also unsuccessful in finding the definition in his notes.

My best guess is that $\operatorname{Proj}_A S$ is in fact $\operatorname{Proj} (S\otimes_\mathbb{Z}A)$ considered in the natural way over $\operatorname{Spec} A$. Is this correct, or is it something else?

Answer 1

The first instance of this text in the most recent edition (November 18, 2017 draft) is in exercise 17.2.A, which begins with the following:

"Suppose ... $S_\bullet$ is a $\Bbb Z^{\geq 0}$-graded ring over $A$."

This means that $S_\bullet$ has the structure of an $A$-algebra, contrary to your assertion. The notation $\operatorname{Proj}_A S_\bullet$ means we're considering $\operatorname{Proj} S_\bullet$ as a scheme over $\operatorname{Spec} A$ - Vakil is preparing you for the relative Proj functor, where it's important to specify the base you're taking Proj of your graded sheaf of modules over.