Scheme-theoretic fibre
= Scheme-theoretic fibre
{title2=$X_y=X\times_Y\operatorname{Spec}\kappa(y)$}
The fibre of a <morphism of schemes> over $y\in Y$ is its base change to the <residue field> $\kappa(y)$. For $\operatorname{Spec}B\to\operatorname{Spec}A$, with $y$ corresponding to $\mathfrak p$, its fibre ring is $B\otimes_A\kappa(\mathfrak p)$. Its <structure sheaf> records nilpotents and multiplicities that the set-theoretic inverse image cannot see.