# A lemma to identify dvrs

In this post I will prove a commutative algebra lemma, which is proved in page 7 of Serre’s Local Fields. This lemma is useful if you find a ring in the wild and want to know that it’s a discrete valuation ring.

Proposition: Let be a noetherian local domain where and is not nilpotent. Then .

Proof: Suppose . Then for all . So for all . Since is a domain, .

Therefore consider the ascending chain . This eventually stabilizes for high enough since is noetherian, so for some , . Thus , so . But is a unit, so , so .

This theorem holds more generally even if is not assumed to be a domain, but the proof is more complicated (but still among the same lines).

Proposition: Let be a noetherian local ring where and is not nilpotent. Then .

Proof: Let be the ideal of elements that kill some power of . We will use variables to refer to elements of . Since is noetherian, must be finitely generated, so all elements of kill for some fixed .

Now suppose . , so . Thus .

Consider the ascending chain . Since is noetherian it must eventually stablize, so for some , can be written as . But recall that . So so . is a unit since , and is local, so . If we force to be large enough to surpass , then , so .

# Hilbert’s Basis Theorem

Here is a proof of Hilbert’s Basis Theorem I thought of last night.

Let be a noetherian ring. Consider an ideal in . Let be the ideal in generated by the leading coefficients of the polynomials of degree in . Notice that , since if , , and it has the same leading coefficient. Thus we have an ascending chain , which must terminate, since is noetherian. Suppose it terminates at , so .

Now for each choose a finite set of generators (which we can do since is noetherian). For each generator, choose a representative polynomial in with that leading coefficient. This gives us a finite collection polynomials: define to be the ideal of generated by these polynomials.

Let . I claim . Assume for the sake of contradiction that there is a polynomial of minimal degree (say ) which is in but . If , there is an element of with the same leading coefficient, so is not in but has degree smaller than : contradiction. If is of degree , then there is an element of of which has the same leading coefficient as . Thus is of degree smaller than but is not in : contradiction.

Thus . Since is therefore finitely generated, this proves is noetherian.