The <Proj construction> has underlying set
$$
\operatorname{Proj}S=\{\mathfrak p:\mathfrak p\text{ is a homogeneous prime ideal and }S_+\nsubseteq\mathfrak p\}.
$$
Its closed sets are $V_+(I)=\{\mathfrak p:I\subseteq\mathfrak p\}$ for homogeneous ideals $I$, and its standard opens are $D_+(f)$. The structure sheaf is characterized by
$$
\mathcal O_{\operatorname{Proj}S}(D_+(f))=(S_f)_0
$$
for homogeneous $f$ of positive degree.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 2 / a / Solution

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For a graded ring $S$, $\operatorname{Proj}S$ consists of homogeneous prime ideals not containing the irrelevant ideal $S_+$. Its standard affine opens satisfy $D_+(f)\cong\operatorname{Spec}(S_f)_0$.


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