Scheme-theoretic fiber
= Scheme-theoretic fiber
For a morphism of schemes $f:X\to Y$ and a point $y\in Y$, the scheme-theoretic fiber is the fiber product $X\times_Y\operatorname{Spec}\kappa(y)$. It remembers multiplicities and nilpotents that the underlying set-theoretic inverse image cannot detect.
Its underlying <fiber> is enriched with local-ring and multiplicity information.