Hyperinflation

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. 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. Template:Harvs 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 countable choice) there is a sequence (f_n)_n∈N of polynomials such that if b_n is the left ideal generated by f₀, ..., f_n−1 then f_n in a\b_n is of minimal degree. It is clear that (deg(f_n))_n∈N 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 {a₀, a₁, ...}. Since R is left-Noetherian, we have that b must be finitely generated; and since the a_n comprise an R-basis, it follows that for a finite amount of them, say {a_i : i < N}, will suffice. So for example, $a_N=\sum _{i<N}{u}_i{a}_i$ some u_i in R. Now consider

$g\triangleq \sum _{i<N}{u}_i{X}^{\mathrm{deg}(f_N)-\mathrm{deg}(f_i)}{f}_i,$

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 f₀, ..., f_N−1. Let $d\triangleq {\max }_i\mathrm{deg}(f_i).$ 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 $f_0^{(k)},\dots ,f_{N^{(k)}-1}^{(k)},$ 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 $a=\sum _j{u}_j{a}_j$ of the coefficients of the f_j Consider $\tilde{h}\triangleq \sum _j{u}_j{X}^{\mathrm{deg}(h)-\mathrm{deg}(f_j)}{f}_j,$ which has the same leading term as h; moreover $\tilde{h}\in {\mathfrak{a}}^{\ast }\ni ̸h$ so $h-\tilde{h}\in \mathfrak{a}\setminus {\mathfrak{a}}^{\ast }$ of degree < deg(h), contradicting minimality.

Case 2: deg(h) = k < d. Then a ∈ b_k so is a left-linear combination $a=\sum _j{u}_j{a}_j^{(k)}$ of the leading coefficients of the $f_j^{(k)}.$ Considering $\tilde{h}\triangleq \sum _j{u}_j{X}^{\mathrm{deg}(h)-\mathrm{deg}(f_j^{(k)})}{f}_j^{(k)},$ 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. First, by induction we see that $R[X_0,X_1,\dots ,X_{n-1}]$ will also be Noetherian. Second, since any affine variety over Rⁿ (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal $\mathfrak{a}\subseteq R[X_0,X_1,\dots ,X_{n-1}]$ 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. Finally, if $\mathcal{𝒜}$ is a finitely-generated R-algebra, then we know that $\mathcal{𝒜}\cong R[X_0,X_1,\dots ,X_{n-1}]/⟨\mathfrak{a}⟩$ (i.e. mod-ing out by relations), where a a set of polynomials. We can assume that a is an ideal and thus is finitely generated. So $\mathcal{𝒜}$ is a free R-algebra (on n generators) generated by finitely many relations $\mathcal{𝒜}\cong R[X_0,X_1,\dots ,X_{n-1}]/⟨p_0,\dots ,p_{N-1}⟩$.

Mizar System

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

References

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