TITLE: Is a scheme Noetherian if its topological space and its stalks are?
QUESTION [11 upvotes]: Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?

REPLY [7 votes]: A further counterexample is Example 2.3 in W. Heinzer, J. Ohm, Locally noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273-284.