5
$\begingroup$

Ravi Vakil definition 5.3.4.

Suppose $X$ is a scheme. If $X$ can be covered by affine open sets Spec($A$) where $A$ is Noetherian, we say that $X$ is a locally Noetherian scheme.

The definition only says "If $X$ can be covered by affine open sets Spec($A$) where $A$ is Noetherian". I wonder if this can imply that every affine open set is Northerian?

$\endgroup$
4
$\begingroup$

Yes, because an affine scheme is quasicompact. So, if $U\subseteq X$ is an affine open subscheme, then $U$ is covered by finitely many spectra of Noetherian rings and so is Noetherian.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.