Lemma 27.12.5. Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. The functor $F$ defined above is representable by the scheme $X$.
Proof. We have seen above that the functor $F_ d$ corresponds to the open subscheme $U_ d \subset X$. Moreover the transformation of functors $F_ d \to F_{d'}$ (if $d | d'$) defined above corresponds to the inclusion morphism $U_ d \to U_{d'}$ (see discussion above). Hence to show that $F$ is represented by $X$ it suffices to show that $T \to X$ for a quasi-compact scheme $T$ ends up in some $U_ d$, and that for a general scheme $T$ we have
\[ \mathop{\mathrm{Mor}}\nolimits (T, X) = \mathop{\mathrm{lim}}\nolimits _{V \subset T\text{ quasi-compact open}} \mathop{\mathrm{Mor}}\nolimits (V, X). \]
These verifications are omitted. $\square$
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