= Fibre of absolute Frobenius over a rational point of the affine line
{c}
{title2=$\operatorname{Spec}k[t]/(t-a)^p$}
Over a <perfect field> $k$ of characteristic $p$, the <scheme-theoretic fiber> of the <Absolute Frobenius morphism> of $\mathbb A_k^1$ over a rational point $a$ is
$$
\operatorname{Spec}\bigl(k[t]\otimes_{k[t],\,t\mapsto t^p}k\bigr)
\cong\operatorname{Spec}k[t]/(t-a)^p.
$$
Its underlying space is one point, but its coordinate ring has a nonzero <nilpotent element>, so the fibre is a length-$p$ <nonreduced scheme>.
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