Question about localisation in Hartshorne

Question

In proposition 2.5 (p77) of Hartshorne's book, Hartshorne considers a graded ring $S = \bigoplus_{d =0}^\infty$ and the localisation $S_f$ of a homogeneous element $f \in S^+= \bigoplus_{d=1}^\infty S_d$ (i.e. $f \in S_d$ for some $d \geq 1$).

I.e. we localise $S$ w.r.t. the multiplicatively closed set $\{1,f,f^2, f^3, \dots\}$. How are we sure this localisation makes sense? I.e. why is $f$ not nilpotent?

Answer 1

Let's recall the relevant portion of the proposition in question:

Let $S$ be a graded ring.

(b) For any homogeneous $f\in S_+$, let $D_+(f) = \{ \mathfrak{p}\in\operatorname{Proj} S \mid f\notin \mathfrak{p}\}$. Then $D_+(f)$ is open in $\operatorname{Proj} S$. Furthermore, these open sets cover $\operatorname{Proj} S$, and for each such open set, we have an isomorphism of locally ringed spaces $(D_+(f),\mathcal{O}|_{D_+(f)}) \cong \operatorname{Spec} S_{(f)}$ where $S_{(f)}$ is the subring of elements of degree zero of the localization $S_f$.

If $f$ is nilpotent, then $S_f=S_{(f)}=0$, and $\operatorname{Spec} S_{(f)} = \emptyset$, which is a perfectly fine open subset.