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

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?

• Please use \operatorname{Proj} to format $\operatorname{Proj}$. I've made this upgrade for you this time. Commented Apr 21, 2021 at 17:52

"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.
• Thanks a lot. That makes much more sense. I was looking at one of the 2013 drafts which does not have the 'over $A$' part. Commented Apr 21, 2021 at 17:57