Is a factorial scheme with Noetherian stalks locally Noetherian?

Question

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.

Answer 1

Here are two negative results in this direction.

A ring whose stalks are noetherian and factorial need not be coherent.

A counterexample was constructed by Harris and Nagata; see Example on p. 51 in S. Glaz, Commutative coherent rings, Lecture Notes in Math. 1371, Springer, Berlin, 1989.

A domain whose stalks are discrete valuation rings need not be noetherian.

A counterexample was constructed by Heinzer and Ohm; see Example 2.2 in W. Heinzer, J. Ohm, Locally noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273-284.