{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>.


 Fibre of absolute Frobenius over a rational point of the affine line

ID: fibre-of-absolute-frobenius-over-a-rational-point-of-the-affine-line