Hilbert's basis theorem

Statement

If R a ring, let R [ X ] denote the ring of polynomials in the indeterminate X over R . Hilbert proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R [ X ] . Formally,

Hilbert's Basis Theorem. If R is a Noetherian ring, then R [ X ] is a Noetherian ring.

Corollary. If R is a Noetherian ring, then R [ X 1 , … , X n ] is a Noetherian ring.

This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. Hilbert (1890) proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.

Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases.

Proof

Theorem. If R is a left (resp. right) Noetherian ring, then the polynomial ring R [ X ] is also a left (resp. right) Noetherian ring.

Remark. We will give two proofs, in both only the "left" case is considered, the proof for the right case is similar.

First Proof

Suppose a ⊆ R [ X ] were a non-finitely generated left-ideal. Then by recursion (using the axiom of dependent choice) there is a sequence { f 0 , f 1 , … } of polynomials such that if b n is the left ideal generated by f 0 , … , f n − 1 then f n in a ∖ b n is of minimal degree. It is clear that { deg ⁡ ( f 0 ) , deg ⁡ ( f 1 ) , … } is a non-decreasing sequence of naturals. Let a n be the leading coefficient of f n and let b be the left ideal in R generated by a 0 , a 1 , … . Since R is Noetherian the chain of ideals ( a 0 ) ⊂ ( a 0 , a 1 ) ⊂ ( a 0 , a 1 , a 2 ) … must terminate. Thus b = ( a 0 , … , a N − 1 ) for some integer N . So in particular,

a N = ∑ i < N u i a i , u i ∈ R .

Now consider

g = ∑ i < N u i X deg ⁡ ( f N ) − deg ⁡ ( f i ) f i ,

whose leading term is equal to that of f N ; moreover, g ∈ b N . However, f N ∉ b N , which means that f N − g ∈ a ∖ b N has degree less than f N , contradicting the minimality.

Second Proof

Let a ⊆ R [ X ] be a left-ideal. Let b be the set of leading coefficients of members of a . This is obviously a left-ideal over R , and so is finitely generated by the leading coefficients of finitely many members of a ; say f 0 , … , f N − 1 . Let d be the maximum of the set { deg ⁡ ( f 0 ) , … , deg ⁡ ( f N − 1 ) } , and let b k be the set of leading coefficients of members of a , whose degree is ≤ k . As before, the b k are left-ideals over R , and so are finitely generated by the leading coefficients of finitely many members of a , say

f 0 ( k ) , ⋯ , f N ( k ) − 1 ( k ) ,

with degrees ≤ k . Now let a ∗ ⊆ R [ X ] be the left-ideal generated by

{ f i , f j ( k )   :   i < N , j < N ( k ) , k < d } .

We have a ∗ ⊆ a and claim also a ⊆ a ∗ . Suppose for the sake of contradiction this is not so. Then let h ∈ a ∖ a ∗ be of minimal degree, and denote its leading coefficient by a .

Case 1: deg ⁡ ( h ) ≥ d . Regardless of this condition, we have a ∈ b , so is a left-linear combination of the coefficients of the f j . Consider which has the same leading term as h ; moreover h 0 ∈ a ∗ while h ∉ a ∗ . Therefore h − h 0 ∈ a ∖ a ∗ and deg ⁡ ( h − h 0 ′ ) < deg ⁡ ( h ) , which contradicts minimality. Case 2: deg ⁡ ( h ) = k < d . Then a ∈ b k so is a left-linear combination of the leading coefficients of the f j ( k ) . Considering we yield a similar contradiction as in Case 1.

Thus our claim holds, and a = a ∗ which is finitely generated.

Note that the only reason we had to split into two cases was to ensure that the powers of X multiplying the factors, were non-negative in the constructions.

Applications

Let R be a Noetherian commutative ring. Hilbert's basis theorem has some immediate corollaries.

1. By induction we see that R [ X 0 , … , X n − 1 ] will also be Noetherian.
2. Since any affine variety over R n (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal a ⊂ R [ X 0 , … , X n − 1 ] and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces.
3. If A is a finitely-generated R -algebra, then we know that A ≃ R [ X 0 , … , X n − 1 ] / a , where a is an ideal. The basis theorem implies that a must be finitely generated, say a = ( p 0 , … , p N − 1 ) , i.e. A is finitely presented.

Mizar System

The Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in the HILBASIS file.