 Generic local ring

In the sheaf topos for the Zariski coverage on finitely presented rings, the universal local ring is represented by the polynomial ring $\mathbb{Z}[t]$. At the stage of a finitely presented ring $A$, its section set is $A$, and restriction along $A\to B$ is that ring map. It is internally local because the localizations at $x$ and $1-x$ cover each stage. Its square-zero infinitesimals are represented by $\mathbb{Z}[\epsilon ]/({\epsilon }^2)$. Stage rings need not themselves be local or reduced; their variation supplies the internal local and infinitesimal semantics.