Generic local ring ($R(A)=A$)

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.

For $D=\{d\in R:d^2=0\}$ in the generic local ring, every internal map $D\to R$ has a unique affine expression $d\mapsto a+bd$. At the stage $A$, such maps are elements of $A[\epsilon ]/({\epsilon }^2)$, which is freely $A\oplus A\epsilon$ as an $A$-module. This decomposition is natural under ring maps and proves the isomorphism. Testing only on ordinary fields loses the generic infinitesimal and would give an incorrect uniqueness argument.