Past exam of the introduction to schemes course of the University of Oxford / 2017 / 2 / a / Solution

The localization of a ring $R_r$ is generated over $R$ by the single element $1/r$, since every fraction can be written asThe presentation of a localization at one element isTo verify the presentation even when $R$ has zero divisors, observe that $r$ is invertible in $R[t]/(rt-1)$ with inverse $t$. Hence the map from $R$ to this quotient ring extends uniquely to $R_r$. This extension and the displayed map are inverse ring homomorphisms: they agree on $R$ and send the designated inverse of $r$ to the designated inverse of $r$.

Thus $R_r$ is a finitely generated algebra over a ring, and the criterion for a morphism of finite type between affine schemes applies to $u:\mathrm{Spec}{R}_r\to \mathrm{Spec}R$. There is just one affine scheme in the required cover of the source, with one algebra generator. Therefore $u$ is of finite type. If $r$ is nilpotent, the same presentation is the zero ring, and the source is the empty scheme; the conclusion still holds.