A
scheme
admitting a finite open covering by spectra of Noetherian rings (cf.
Noetherian ring).
An affine Noetherian scheme is precisely the spectrum of
a Noetherian ring. The topological space of a Noetherian scheme
is a Noetherian topological space, and the local rings
are Noetherian. If every point of a scheme has
an open affine Noetherian neighbourhood, the scheme is called
locally Noetherian.
A quasi-compact locally Noetherian scheme is a Noetherian scheme. An example of
a Noetherian scheme is a scheme of finite type over
a field (an algebraic variety) or over any Noetherian ring.