I briefly outlined the definition and first properties of Noetherian rings and modules a while back.  There are several useful and well-known criteria to tell whether a ring is Noetherian, as I will discuss in this post.  Actually, I’ll only get to the first few basic ones here, though these alone give us a lot of tools for, say, algebraic geometry, when we want to show our schemes are relatively well-behaved.  But there are plenty more to go.

 Hilbert’s basis theorem

 It is the following: 

Theorem 1 (Hilbert) Let {A} be a Noetherian ring. Then the polynomial ring {A[X]} is also Noetherian.

 Pick an ideal {I \subset A[X]}; we must show {I} is finitely generated.

For each {n}, consider the set of all polynomials in {I} of degree {n}, and let {J_n} denote the set of their leading coefficients. Now {J_n} is actually an ideal in {A} because {I=AI}; note also that {J_n \subset J_m} if {n<m}, because one can multiply an element of {I} by {X^{m-n}} and stay in {I}. Since {A} is Noetherian, the {J_n} eventually stabilize at some {J}, which must be finitely generated. Let {a_1, \dots, a_k} be generators; by definition, these are the leading coefficients of some polynomials {P_1, \dots, P_k \in I}. Set {d := \mathrm{max}(deg \ P_i)}.

The idea now is that given any polynomial {P \in I}, we can subtract multiples of {P_1, \dots, P_k} to bring the degree down to less than {d}. Suppose {D := deg \ P > d}. Then the leading coefficient {p_0} of {P} lies in {J}, say

\displaystyle p_0 = c_1 a_1 + \dots c_k a_k;

then

\displaystyle P - \sum c_i X^{D - deg \ P_i} P_i \in I

has degree strictly less than that of {P}, and we proceed inductively.

So the {P_i}‘s aren’t yet generators of {I}, but almost there: we just have to find {A}-generators of the {A}-module {M := \{ Q \in I: deg \ Q < d \}}, and we can pool these with the {P_i}‘s to get {A[X]}-generators of {I}. But {M} is a submodule of the finitely generated {A}-module {A^d}, hence finitely generated since {A} is Noetherian. This completes the proof.

The basis theorem is actually often used in the following form: 

Corollary 2 Let {A} be a Noetherian ring and {B \supset A} a finitely generated {A}-algebra. Then {B} is Noetherian.

 Since the quotient ring of a Noetherian ring is Noetherian, we reduce to {B} a polynomial ring; then it follows from Hilbert’s theorem and induction.

In algebraic geometry, this is important because it implies for instance that a scheme of finite type over a field is Noetherian, and is thus fairly well-behaved. As another example, it shows that a subvariety of {A^n_k}—which by definition is defined by the zero set of some ideal in the polynomial ring {k[X_1, \dots, X_n]}—can be cut out by a finite number of polynomial equations: just take generators of that ideal.

 Localization Criteria

 Localization preserves the property of a ring being Noetherian: 

Proposition 3 If {A} is a Noetherian ring and {S} a multiplicative subset, then {S^{-1}A} is a Noetherian ring.

 This follows from general facts about localization, namely that any ideal of {S^{-1}A} is of the form {S^{-1}I} for {I \subset A} an ideal; hence since {I} is finitely generated, so is {S^{-1}I}.

There is a more interesting converse: If sufficiently many localizations are Noetherian, so is the original ring. 

Theorem 4 Let {A} be a ring with elements {f_1, \dots, f_n} generating the unit ideal. If each localization {A_{f_i}} is Noetherian, so is {A}.

 

For each {i}, there are localization maps {\phi_i: A \rightarrow A_{f_i}}. Given an ideal {I \subset A}, we associate the ideals {I_{f_i} \subset A_{f_i}}. By definition, these are defined as {\phi_i(I) A_{f_i}}.

I claim that

\displaystyle I = \bigcap_i \phi_i^{-1}( I_{f_i} ).\ \ \ \ \ (1)

Indeed, the inclusion {\subset} is clear from the definitions, but the other inclusion requires more checking. Suppose {x \in A} is such that {\phi_i(x) \in I_{f_i}} for all {i}; this means that there are powers {f_i^{m_i}} with

\displaystyle f_i^{m_i} x \in I.

So, consider the ideal {J := \{ y\in A: yx \in I \}}; then {f_i^{m_i} \in J}. If we show that {J=(1)}, then {1x = x \in I}, and my claim will be proved.

We use: 

Lemma 5 If {f_1, \dots, f_n} generate the unit ideal in {A}, so do {f_1^m, \dots, f_n^m} for any {m}.

Indeed, given a relation

\displaystyle \sum d_i f_i = 1 ,

raise it to a high power {M} to get

\displaystyle \left ( \sum d_i f_i \right)^M = 1,

and every term on the left, when expanded, will lie in the ideal generated by {f_1^m \dots, f_n^m} if {M >> m} is very large.

Return to the proof of the theorem. If we have a sequence {I^{(1)}, I^{(2)}, \dots} of ideals, then for each {i}, the {I^{(j)}_{f_i}} each stabilize since {A_{f_i}} is Noetherian; thus by (1), so do the {I^{(j)}}.

This result implies the following (see e.g. Hartshorne, II.3): 

Corollary 6 If {\mathrm{Spec} \ A} is a locally Noetherian scheme, then {A} is a Noetherian ring.

 This is interesting because being locally Noetherian says something about an open affine cover via Noetherian rings.  But this result says that any open affine subscheme comes from a Noetherian ring.

So, we got to some of the basic criteria.  But there are other questions that arise.  For instance, is a subring of a Noetherian ring Noetherian?  In general, no, but there are important cases when we can say yes.   I’ll discuss this in the next post.

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