Considering it has been at least a post removed, I’ll bring us back to our situation. We have a local Noetherian ring . Our notation is that is the least number of generators of an -primary ideal (which was shown to be independent of choice of ideal here). The goal for the day is to show that .
Suppose is -primary. We’ll prove something more general. Let be a finitely generated -module, a non-zero divisor in and . Then the claim is that .
Since is not a zero-divisor, we have an iso as -modules: . Define . Now take . Since is a stable -filtration of , by Artin-Rees we get that is a stable -filtration of .
For each we have exact.
Thus we get . So if we let , we get for large : .
But is also a stable -filtration of , since . We already showed that the degree and leading coefficient of depends only on and and not on the filtration. Thus and have the same degree and leading coefficient, so the highest powers kill eachother which gives .
In particular, we will need that as an -module gives us .
Now we prove the goal for today. For simplicity, let . We will induct on . The base case gives that is constant for large . In particular, there is some such that for all . But we are local, so and hence by Nakayama, . Thus for any prime ideal , we have for some , so take radicals to get . Thus there is only one prime ideal and we actually have an Artinian ring and hence have .
Now suppose and the result holds for . Let be a chain of primes. Choose . Define and be the image of in .
Note that since is an integral domain, and is not 0, it is not a zero-divisor. So we use our first proof from today to get that .
Let be the maximal ideal of . Then is the image of , so which is precisely . Plugging this into the above inequality gives .
So by the inductive hypothesis, . Take our original prime chain. The images form a chain in . Thus . Since the chain was arbitrary, .
A nice corollary here is that the dimension of any Noetherian local ring is finite. Another similar corollary is that in any Noetherian ring (drop the local) the height of a prime ideal is finite (and hence primes satisfy the DCC), since which is local Noetherian.