= Generic local ring
{title2=$R(A)=A$}
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 $\mathbb Z[\varepsilon]/(\varepsilon^2)$. Stage rings need not themselves be local or reduced; their variation supplies the internal local and infinitesimal semantics.
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