I have read in many places that the noetherian hypothesis is often overkill – both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite presentation are the really important properties. This is nicely stated in the foreword to Quitté and Lombardi's Commutative Algebra – Constructive Methods. An excerpt from it appears in Darij Grinberg's answer to this question. I have included it below. Martin Brandenburg's comment on the same question mentions some results true "mainly" for Noetherian rings, e.g $\dim(R[T])=\dim(R)+1$. In this question, an elementary characterization of Krull dimension by Lombardi and Coquand is mentioned in hopes it would provide a simpler proof. Since noetherianity is non-constructive, perhaps it is actually overkill here as well…
My problems are I don't know any examples, don't have enough time to dive into the proof of every result involving Noetherianity, and don't have any intuition to feel when it's overkill.
So I'm looking for nice examples of facts which are often stated for Noetherian rings/schemes but really only require, say, (quasi)coherence. If that's too optimistic, maybe some "right theorems" in the sense of the excerpt.
Finally, let us mention two striking traits of this work compared to
classical works on commutative algebra.The first is to have left
Noetherianity on the backburner. Experience shows that indeed
Noetherianity is often too strong an assumption, which hides the true
algorithmic nature of things. For example, such a theorem usually
stated for Noetherian rings and finitely generated modules, when its
proof is examined to extract an algorithm, turns out to be a theorem
on coherent rings and finitely presented modules. The usual theorem is
but a corollary of the right theorem, but with two nonconstructive
arguments allowing to deduce coherence and finite presentation from
Noetherianity and finite generation in classical mathematics. A proof
in the more satisfying framework of coherence and finitely presented
modules is often already published in research articles, although
rarely in an entirely constructive form, but “the right statement” is
generally missing in the reference works.
Best Answer
1) It is sometimes stated that a finitely generated module $M$ over a Noetherian commutative ring $R$ is projective if for all maximal ideals $\mathfrak m\subset R$ the localized module $M_\mathfrak m$ is free over $R_\mathfrak m$.
However the noetherianity of $R$ is unnecessary if you add the hypothesis that $M$ is finitely presented over $R$.
2) Similarly, is a finitely generated flat module $N$ over $R$ projective?
The answer is yes if $R$ is noetherian, but noetherianity is not necessary if you know that $R$ is an integral domain or if you know that $N$ is finitely presented.
3) Finally let me mention Kaplansky's extraordinary theorem:
The ring doesn't need to be noetherian (nor commutative!) and the module needn't even be finitely generated!