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

The Proj construction has underlying setIts 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 byfor homogeneous $f$ of positive degree.

Solved by gpt-5.6-sol high.

A closed immersion identifies its source homeomorphically with a closed subset and induces a surjection from the target structure sheaf to the pushed-forward source structure sheaf.

The quotient map $S\to S/I$ identifies $\mathrm{Proj}(S/I)$ with $V_{+}(I)\subseteq \mathrm{Proj}S$. On every standard open $D_{+}(f)$ it induces the surjectionwith kernel $(I_f{)}_0$. These affine-local maps glue, proving that the natural morphism $\mathrm{Proj}(S/I)\to \mathrm{Proj}S$ is a closed immersion.

Solved by gpt-5.6-sol high.

It suffices to compare the localized homogeneous ideals on every $D_{+}(f)$. If $a\in I$ is homogeneous, choose $N$ so large that $\mathrm{deg}(f^{N}a)\ge n_0$. Then $f^{N}a\in I^{\prime }$, and in $S_f$,Thus $I_f\subseteq I_f^{\prime }$, while the reverse inclusion follows from $I^{\prime }\subseteq I$. Hence $I_f=I_f^{\prime }$, so $(I_f{)}_0=(I_f^{\prime }{)}_0$ and the quotient structure sheaves agree on all standard opens. The two quotients define the same closed subscheme of $\mathrm{Proj}S$.

Solved by gpt-5.6-sol high.