Hilbert's basis theorem

Summary

In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.

StatementEdit

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

Hilbert's Basis Theorem. If   is a Noetherian ring, then   is a Noetherian ring.

Corollary. If   is a Noetherian ring, then   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 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.[1]

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.

ProofEdit

Theorem. If   is a left (resp. right) Noetherian ring, then the polynomial ring   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 ProofEdit

Suppose   is a non-finitely generated left ideal. Then by recursion (using the axiom of dependent choice) there is a sequence of polynomials   such that if   is the left ideal generated by   then   is of minimal degree. It is clear that   is a non-decreasing sequence of natural numbers. Let   be the leading coefficient of   and let   be the left ideal in   generated by  . Since   is Noetherian the chain of ideals

 

must terminate. Thus   for some integer  . So in particular,

 

Now consider

 

whose leading term is equal to that of  ; moreover,  . However,  , which means that   has degree less than  , contradicting the minimality.

Second ProofEdit

Let   be a left ideal. Let   be the set of leading coefficients of members of  . This is obviously a left ideal over  , and so is finitely generated by the leading coefficients of finitely many members of  ; say  . Let   be the maximum of the set  , and let   be the set of leading coefficients of members of  , whose degree is  . As before, the   are left ideals over  , and so are finitely generated by the leading coefficients of finitely many members of  , say

 

with degrees  . Now let   be the left ideal generated by:

 

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

Case 1:  . Regardless of this condition, we have  , so is a left linear combination
 
of the coefficients of the  . Consider
 
which has the same leading term as  ; moreover   while  . Therefore   and  , which contradicts minimality.
Case 2:  . Then   so is a left linear combination
 
of the leading coefficients of the  . Considering
 
we yield a similar contradiction as in Case 1.

Thus our claim holds, and   which is finitely generated.

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

ApplicationsEdit

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

  1. By induction we see that   will also be Noetherian.
  2. Since any affine variety over   (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal   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   is a finitely-generated  -algebra, then we know that  , where   is an ideal. The basis theorem implies that   must be finitely generated, say  , i.e.   is finitely presented.

Formal proofsEdit

Formal proofs of Hilbert's basis theorem have been verified through the Mizar project (see HILBASIS file) and Lean (see ring_theory.polynomial).

ReferencesEdit

  1. ^ Hilbert, David (1890). "Ueber die Theorie der algebraischen Formen". Mathematische Annalen. 36 (4): 473–534. doi:10.1007/BF01208503. ISSN 0025-5831. S2CID 179177713.

Further readingEdit

  • Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.