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

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 $f:U\to \mathrm{Proj}S$.