Hilbert's basis theorem

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. 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

The following more general statement will be proved.

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

It suffices to consider just the "Left" case.

Proof (Theorem)

Suppose per contra that $\mathfrak{a}\subseteq R[X]\,$ were a non-finitely generated left-ideal. Then it would be that by recursion (using the axiom of countable choice) that a sequence $(f_{n})_{n\in\mathbb{N}}\,$ of polynomials could be found so that, letting $\mathfrak{b}_{n}\triangleq\langle f_{0},\ldots,f_{n-1}\rangle,\,$ $f_{n}\in \mathfrak{a}\setminus \mathfrak{b}_{n}\,$ of minimal degree. It is clear that $(\deg(f_{n}))_{n\in\mathbb{N}}\,$ is a non-decreasing sequence of naturals. Now consider the left-ideal $\mathfrak{b}\triangleq\langle a_{n}:n\in\mathbb{N}\rangle\,$ over $R,\,$ where the $a_{n}\,$ are the leading coefficients of the $f_{n}.\,$. Since $R\,$ is left-Noetherian, we have that $\mathfrak{b}\,$ 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, $a_{N}=\Sigma_{i<N}u_{i}a_{i}\,$ some $u_{i}\in R.\,$ Now consider $\tilde{f}(X)\triangleq\Sigma_{i<N}u_{i}X^{\deg(f_{N})-\deg(f_{i})}f_{i}(X),\,$ whose leading term is equal to that of $f_{N}\,;$ moreover $\tilde{f}\in\mathfrak{b}_{N}\not\ni f_{N}\,$ so $f_{N}-\tilde{f}\in\mathfrak{a}\setminus\mathfrak{b}_{N}\,$ of degree $<\deg(f_{N}),\,$ contradicting minimality.

$\blacksquare\,$ (Thm)

A constructive proof (not invoking the axiom of choice) also exists. However, the proof must use Zorn's Lemma which is equivalent to the axiom of choice.

Proof (Theorem):

Let $\mathfrak{a}\subseteq R[X]\,$ be a left-ideal. Let $\mathfrak{b}\,$ be the set of leading coefficients of members of $\mathfrak{a}.\,$ This is obviously a left-ideal over $R,\,$ and so is finitely generated by the leading coefficients of finitely many members of $\mathfrak{a};\,$ say $f_{0},\ldots,f_{N-1}.\,$ Let $d\triangleq\max_{i}\deg(f_{i}).\,$ Let $\mathfrak{b}_{k}\,$ be the set of leading coefficients of members of $\mathfrak{a},\,$ whose degree is $\leq k.\,$ As before, the $\mathfrak{b}_{k}\,$ are left-ideals over $R,\,$ and so are finitely generated by the leading coefficients of finitely many members of $\mathfrak{a},\,$ say $f^{(k)}_{0},\ldots,f^{(k)}_{N^{(k)}-1},\,$ with degrees $\leq k.\,$ Now let $\mathfrak{a}^{\ast}\subseteq R[X]\,$ be the left-ideal generated by $\{f_{i},f^{(k)}_{j}:i<N,j<N^{(k)},k<d\}.\,$ We have $\mathfrak{a}^{\ast}\subseteq\mathfrak{a}\,$ and claim also $\mathfrak{a}\subseteq\mathfrak{a}^{\ast}.\,$

Suppose per contra this were not so. Then let $h\in\mathfrak{a}\setminus\mathfrak{a}^{\ast}\,$ be of minimal degree, and denote its leading coefficient by $a.\,$

Case 1: $\deg(h)\geq d.\,$ Regardless of this condition, we have $a\in \mathfrak{b},\,$ so is a left-linear combination $a=\Sigma_{j}u_{j}a_{j}\,$ of the coefficients of the $f_{j}.\,$ Consider $\tilde{h}\triangleq\Sigma_{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j},\,$ which has the same leading term as h; moreover $\tilde{h}\in\mathfrak{a}^{\ast}\not\ni h\,$ so $h-\tilde{h}\in\mathfrak{a}\setminus \mathfrak{a}^{\ast}\,$ of degree $<\deg(h),\,$ contradicting minimality.

Case 2: $\deg(h)=k<d.\,$ Then $a\in \mathfrak{b}_{k}\,$ so is a left-linear combination $a=\Sigma_{j}u_{j}a^{(k)}_{j}\,$ of the leading coefficients of the $f^{(k)}_{j}.\,$ Considering $\tilde{h}\triangleq\Sigma_{j}u_{j}X^{\deg(h)-\deg(f^{(k)}_{j})}f^{(k)}_{j},\,$ we yield a similar contradiction as in Case 1.

Thus our claim holds, and $\mathfrak{a}=\mathfrak{a}^{\ast}\,$ which is finitely generated.

$\blacksquare\,$ (Thm)

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 Nötherian commutative ring. Hilbert's basis theorem has some immediate corollaries. First, by induction we see that $R[X_{0},X_{1},\ldots,X_{n-1}]\,$ will also be Nötherian. Second, 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 $\mathfrak{a}\subseteq R[X_{0},X_{1},\ldots,X_{n-1}]\,$ 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 $\mathcal{A}\,$ were a finitely-generated $R\,$-algebra, then we know that $\mathcal{A}\cong R[X_{0},X_{1},\ldots,X_{n-1}]/\langle\mathfrak{a}\rangle\,$ (i.e. mod-ing out by relations), where $\mathfrak{a}\,$ a set of polynomials. We can assume that $\mathfrak{a}\,$ is an ideal and thus is finitely generated. So $\mathcal{A}\,$ would be a free $R\,$-algebra (on $n\,$ generators) generated by finitely many relations $\mathcal{A}\cong R[X_{0},X_{1},\ldots,X_{n-1}]/\langle p_{0},\ldots,p_{N-1}\rangle$.

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