Modern algebraic geometry makes heavy use of two constructions that turn rings into schemes: Spec and Proj. The first construction turns a commutative ring $A$ into a scheme $\operatorname{Spec}(A)$, called the "prime spectrum" of $A$. The second construction turns a graded ring $S$ into a scheme $\operatorname{Proj}(S)$, called $\dots$ "Proj of $S$", according to these three standard sources:
- The Stacks Project (http://stacks.math.columbia.edu/tag/00JM)
- Vakil's The Rising Sea (section 4.5)
- Hartshorne's Algebraic Geometry (section II.2)
Usually the name of a mathematical object precedes its notation, but for $\operatorname{Proj}(S)$, it seems like the name is the notation. Is that right? What is $\operatorname{Proj}(S)$ actually called?
In the interest of staying within the scope of MSE and avoiding personal opinions, I'd like a reference to a source that gives $\operatorname{Proj}(S)$ a name.
After writing this question, I dug a bit deeper and looked at (where else?) EGA. Section 2.3 of Chapter II calls $\operatorname{Proj}(S)$ the spectre premier homogène, which I would translate as homogeneous prime spectrum. In retrospect, this name is very reasonable; I'm not sure why it isn't more widely used. (Googling "prime spectrum" gives 20,400 hits but "homogeneous prime spectrum" gives only about 500.) It's probably just easier to say "Proj of $S$."