# Is a factorial scheme with Noetherian stalks locally Noetherian?

Motivation: The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct sums, we obtain that sheaves of sections of holomorphic vector bundles (i.e. locally free sheaves of modules of finite rank) are coherent, as well. It seems that any reasonable proof of the Oka's theorem extensively uses that the local ring of the structure sheaf of a complex manifold is a Noetherian UFD.

I am thinking about an anology in algebraic geometry. The most reasonable property of a scheme that makes its structure sheaf coherent is local Noetherianity. So, suppose we have a scheme whose local rings are Noetherian UFDs. Is it true that it will be locally Noetherian? In other words, is it true that a ring whose localizations at all primes are Noetherian UFDs is Noetherian? If not, is there still any hope for its structure sheaf to be coherent?

Thank you a lot.

• I think the example by Martin Brandenburg in this answer is also a counterexample to your question. Dec 5, 2018 at 14:59