nLab locally algebraic scheme

A scheme over a field $k$ is called locally algebraic scheme if it has a covering $\{X_i\}$ of (open affine sub functors which are) affine schemes and all $O(X_i)$ are finitely generated $k$-algebras, i.e. if it is locally of finite type over $Spec k$.

A scheme is called algebraic scheme if moreover there exist a finite such covering, i.e. if it is of finite type over $Spec k$.

category: algebraic geometry