In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to $\textrm{Spec } S_{(f)}$, the latter denoting the degree zero elements in the localization $S_f$.
In all constructions I have seen of this, including Hartshorne and Liu, they usually leave it at that with no mention of global sections. I was tempted to handle this by saying $f = 1$ then $D_+ (f) = \textrm{Proj } S$ is isomorphic to $\textrm{Spec } S_{(1)} = \textrm{Spec } S_0$ (degree 0 elements of $S$). The Wikipedia article mentions this without proof or justification under the "Twisting Sheaf of Serre" section (https://en.wikipedia.org/wiki/Proj_construction#The_twisting_sheaf_of_Serre) lending credence to this fact.
However, the condition that $f \in S_+$, where $1$ typically does not reside (take the natural grading on the polynomial ring for example) seems to invalidate this proof. What justification should I use instead?
Background: I am interested in justifying that for $X = \textrm{Proj } A[x_0,\dots,x_n]$, that $\mathcal{O}_X (1)$ can be generated by the global sections $x_0, \dots, x_n$.
Best Answer
Let $A$ be a commutative unital ring and let $E:=A\{e_0,..,e_n\}$ be the free $A$-module of rank $n+1$ on the elements $e_i$. Let $E^*:=A\{x_0,..,x_n \}$ be the dual of $E$ and let $S:=Sym_A(E^*)\cong A[x_0,..,x_n]$. Let $\mathbb{P}(\mathcal{E}^*):=Proj(S)$ be the projective space bundle of $\mathcal{E}$ (the sheafification of $E$). There is a canonical exact sequence of graded $S$-modules
$\phi: \oplus_{i=0}^n S x_i \rightarrow S(1) \rightarrow 0$
defined by
$\phi( f_0,f_1,..,f_n):= \sum_{i=0}^n f_i x_i .$
Let $\pi:\mathbb{P}(\mathcal{E}^*) \rightarrow Y$ (here $Y:=Spec(A)$) be the projection map. When sheafifying the map $\phi$ we get the tautological sequence
$\pi^*\mathcal{E}^* \rightarrow^{\phi} \mathcal{O}(1) \rightarrow 0.$
Since $\pi^*\mathcal{E}^*$ is a free $\mathcal{O}$-module it follows this proves the invertible sheaf $\mathcal{O}(1)$ is generated by the global sections $s_i:=\phi(x_i)$.
Lemma: Let $X$ be a scheme. For a surjection $\phi: \mathcal{O}_X \{y_1,..,y_l\} \rightarrow \mathcal{L}$ where $\mathcal{L}$ is an invertible sheaf on $X$ and $\mathcal{O}_X\{y_1,..,y_l\}$ is the free sheaf of rank $l$ on the elements $y_i$, it follows $\phi(y_i)$ are global sections generating $\mathcal{L}$. Conversely given a set of global sections $s_1,..,s_l$ of $\mathcal{L}$ that generated $\mathcal{L}$ it follows there is a canonical surjective map of $\mathcal{O}_X$-modules $\phi: \mathcal{O}_X\{y_1,..,y_l\}\rightarrow \mathcal{L}$ defined on an open set $U$ by $\phi(\sum_i u_iy_i):=\sum_i u_i(s_i)_U$ where $(s_i)_U$ is the restriction of $s_i$ to $U$. The details may be found in Hartshorne's book Chapter II.7.