 Locally Noetherian scheme

A scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings. It is a Noetherian scheme exactly when it is also a quasi-compact scheme: a finite subcover then supplies the required finite affine open cover. An infinite gluing of spectra of Noetherian rings along open subschemes is still locally Noetherian, but need not be a quasi-compact scheme.