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authorGravatar Tim Hosgood <timhosgood@gmail.com> 2021-04-23 17:17:47 +0100
committerGravatar Tim Hosgood <timhosgood@gmail.com> 2021-04-23 17:17:47 +0100
commit74a9e5dc9ad219b1d5432f5790750b6e5b12a6f5 (patch)
treeae69260c00c24ec155e2b0773eb36a94994186d6
parent2b10710d5a3c5e8c8c4a3140e8e5b0070d409c23 (diff)
downloadega2-2.tar.gz
ega2-2.zip
starting II.2.7ega2-2
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@@ -1149,3 +1149,21 @@ over $D_+(f)$, i.e. the restriction of $z$ to $D_+(f)$, which finishes the proof
\subsection{Finiteness conditions}
\label{subsection:II.2.7}
+
+\begin{proposition}[2.7.1]
+\label{II.2.7.1}
+\begin{enumerate}
+ \item[{\rm(i)}] If $S$ is a graded Noetherian ring, then $X=\Proj(S)$ is a Noetherian scheme.
+ \item[{\rm(ii)}] If $S$ is a graded $A$-algebra of finite type, then $X=\Proj(S)$ is a scheme of finite type over $Y=\Spec(A)$.
+\end{enumerate}
+\end{proposition}
+
+\oldpage[II]{39}
+\begin{proof}
+\begin{enumerate}
+ \item[{\rm(i)}] If $S$ is Noetherian, then the ideal $S_+$ admits a finite system of homogeneous generators $f_i$ ($1\leq i\leq p$), thus \sref{II.2.3.14} the underlying space $X$ is the union of the $D_+(f_i)=\Spec(S_{(f_i)})$, and everything then reduces to showing that each of the $S_{(f_i)}$ is Noetherian, which follows from \sref{II.2.2.6}.
+ \item[{\rm(ii)}] The hypothesis implies that $S_0$ is an $A$-algebra of finite type, and that $S$ is an $S_0$-algebra of finite type, and so $S_+$ is an ideal of finite type \sref{II.2.1.4}.
+ We are thus reduced, as in (i), to showing that $S_{(f)}$ is an $A$-algebra of finite type for all $f\in S_d$.
+ By \sref{II.2.2.5}, it suffices to show that $S^{(d)}$ is an $A$-algebra of finite type, which follows from \sref{II.2.1.6}.
+\end{enumerate}
+\end{proof}