Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 113 / 1 / c / Solution

Because ${\mathcal{A}}_{\bullet }$ is coherent and locally generated by ${\mathcal{A}}_1$, there is locally a finite-rank coherent sheaf $\mathcal{E}$ and a surjection of graded algebrasThe Relative Proj construction turns this into a closed immersionConsequently $Y\to X$ is a projective morphism and therefore a proper morphism. Since $X\to \mathrm{Spec}k$ is proper and proper morphisms are closed under composition, $Y\to \mathrm{Spec}k$ is proper.