# Examples of Noetherian overkill

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.

• I think the use of quotes around "the right statement" by Lombardi and Quitté is the key. Classical concepts like "noetherian" break down into several concepts that are classically equivalent but constructively inequivalent. So "the right statement" is not a well-defined thing, it's often a dozen different statements, each of which plays a different role in the constructive theory. To find "the right statement" is not a local problem and there is no easy substitute to a global constructive development such as that of Lombardi and Quitté. – François G. Dorais Nov 22 '15 at 13:03
• (Continued:) That said, the "proof mining program" has been very successful at extracting constructive concepts and algorithms from classical theorems. This can be done "locally" to a certain extent but the outcome will not generally be the same as a full constructive development. – François G. Dorais Nov 22 '15 at 13:06
• Main theorems in Northcott's book, Finite Free Resolutions, are based on his theory of grade (Chap. 5). This way he avoided any Noetherianity hypothesis . – Al-Amrani Nov 24 '15 at 12:28

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:

Every projective module over a local ring is free

The ring doesn't need to be noetherian (nor commutative!) and the module needn't even be finitely generated!

• More famous is : a surjecive endomorphism of finitely generated module, over any commutative ring , is an automorphism. (Of course, for Nakayama's Lemma, no noetherianity is necessary .) – Al-Amrani Nov 24 '15 at 9:31
• You write "More famous is ...". On the totally ordered set of mathematical results by famousness, I suppose? And by the way, I was not totally unaware of that result you evoke... – Georges Elencwajg Nov 24 '15 at 18:34