Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 1

For every homogeneous $s\in S_{+}$,Henceso $U$ is open in the Zariski topology.

If $\mathfrak{p}\in U$, then ${\phi }^{-1}(\mathfrak{p})$ is a homogeneous prime ideal that does not contain the irrelevant ideal of a graded ring $S_{+}$. Thusdefines a map $U\to \mathrm{Proj}S$. On every Standard affine open of Proj the graded homomorphism inducesand therefore an affine scheme morphismThese morphisms agree after localization on overlaps, so they glue to the required morphism of schemes $f:U\to \mathrm{Proj}S$.

Let $\mathfrak{p}\in \mathrm{Proj}T$. Some homogeneous $t\in T_{+}$ lies outside $\mathfrak{p}$. Choose $N$ with $N\mathrm{deg}t\ge d_0$. Since ${\phi }_{N\mathrm{deg}t}$ is surjective, $t^N=\phi (s)$ for some homogeneous $s\in S_{+}$; primality gives $t^N\notin \mathfrak{p}$. Thus $\mathfrak{p}\in U$, and $U=\mathrm{Proj}T$.

Fix a positive-degree homogeneous $s\in S$ and write $q=\phi (s)$. The mapis surjective: after multiplying the numerator and denominator of any degree-zero fraction by a sufficiently large power of $q$, its numerator has degree at least $d_0$ and therefore lifts through $\phi$. It is injective by the same device: if $a/s^n$ maps to zero, then $q^m\phi (a)=0$ for some $m$, and after increasing $m$ the injectivity of $\phi$ in degree $(m+n)\mathrm{deg}s$ gives $s^{m}a=0$. Hence the displayed map is a ring isomorphism.

The opens $D_{+}(q)$ obtained in this way cover $\mathrm{Proj}T$, and on every one of them $f$ is an isomorphism onto $D_{+}(s)$. The inverses agree on overlaps, so $f$ is an isomorphism of schemes. This proves the Invariance of Proj under an eventual graded isomorphism.