Noetherian scheme

Remarks

A scheme is locally Noetherian if and only if $A$ is Noetherian for every open affine subscheme $\operatorname{Spec} A$. So $\operatorname{Spec} A$ is Noetherian if and only if $A$ is a Noetherian ring. The underlying space of a Noetherian scheme is a Noetherian topological space. Example: projective space $\mathbb{P}^n_k$ over a field $k$ is Noetherian, being covered by $n+1$ copies of $\operatorname{Spec} k[x_1, \ldots, x_n]$, which are Noetherian by the Hilbert basis theorem. Non-example: $\operatorname{Spec} k[x_1, x_2, \ldots]$ is not Noetherian, since the ideal $(x_1, x_2, \ldots)$ is not finitely generated. Non-example: a disjoint union of infinitely many copies of $\operatorname{Spec} k$ is locally Noetherian but not Noetherian, since it is not quasi-compact.