Locally of finite presentation - Imported Knowlpedia Content

A morphism $f:Y\to X$ of schemes is locally of finite presentation if $X$ has an affine open cover $U_i=\operatorname{Spec}A_i$ such that each $f^{-1}(U_i)$ has an affine open cover $V_{ij}=\operatorname{Spec}B_{ij}$ for which every $A_i$ -algebra $B_{ij}$ is finitely presented.

Explicitly, finitely presented means that for some finite integers $m,n$ ,

B_{ij}\cong A_i[x_1,\ldots,x_n]/(r_1,\ldots,r_m).

This criterion is local on both source and target; the resulting property is independent of the chosen affine covers.

For a field extension $K/F$ , $\operatorname{Spec}K\to\operatorname{Spec}F$ is locally of finite presentation exactly when $K$ is a finitely presented $F$ -algebra. In particular, every finite field extension has this property.