Locally of finite type - Imported Knowlpedia Content

A morphism $f:Y\to X$ of schemes is locally of finite type 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 generated.

Explicitly, finitely generated means that for some finite integer $n$ , there is a surjective $A_i$ -algebra homomorphism

A_i[x_1,\ldots,x_n]\longrightarrow B_{ij}.

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

Every locally finitely presented morphism is locally of finite type. The converse need not hold, because finite type requires finitely many algebra generators but does not require the ideal of relations among them to be finitely generated.