Hilbert's basis theorem

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

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 $R[X_{1},\dotsc ,X_{n}]$ 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 ${\mathfrak a}\subseteq R[X]$ were a non-finitely generated left-ideal. Then by recursion (using the axiom of dependent choice) there is a sequence $\{f_{0},f_{1},\dotsc \}$ of polynomials such that if ${\mathfrak b}_{n}$ is the left ideal generated by $f_{0},\dotsc ,f_{{n-1}}$ then $f_{n}$ in ${\mathfrak a}\setminus {\mathfrak b}_{n}$ is of minimal degree. It is clear that $\{\deg(f_{0}),\deg(f_{1}),\dotsc \}$ is a non-decreasing sequence of naturals. Let $a_{n}$ be the leading coefficient of $f_{n}$ and let ${\mathfrak b}$ be the left ideal in $R$ generated by $a_{0},a_{1},\dotsc$ . Since $R$ is Noetherian the chain of ideals $(a_{0})\subset (a_{0},a_{1})\subset (a_{0},a_{1},a_{2})\dotsc$ must terminate. Thus ${\mathfrak b}=(a_{0},\dotsc ,a_{{N-1}})$ for some integer $N$ . So in particular,

a_{N}=\sum _{{i<N}}u_{{i}}a_{{i}},\qquad u_{i}\in R.

Now consider

g=\sum _{{i<N}}u_{{i}}X^{{\deg(f_{{N}})-\deg(f_{{i}})}}f_{{i}},

whose leading term is equal to that of $f_{N}$ ; moreover, $g\in {\mathfrak b}_{N}$ . However, $f_{N}\notin {\mathfrak b}_{N}$ , which means that $f_{N}-g\in {\mathfrak a}\setminus {\mathfrak b}_{N}$ has degree less than $f_{N}$ , contradicting the minimality.

Second Proof

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 $f_{0},\dotsc ,f_{{N-1}}$ . Let $d$ be the maximum of the set $\{\deg(f_{0}),\dotsc ,\deg(f_{{N-1}})\}$ , and 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_{{0}}^{{(k)}},\cdots ,f_{{N^{{(k)}}-1}}^{{(k)}},

with degrees ${}\leq k$ . Now let ${\mathfrak a}^{*}\subseteq R[X]$ be the left-ideal generated by

\left\{f_{{i}},f_{{j}}^{{(k)}}\ :\ i<N,j<N^{{(k)}},k<d\right\}.

We have ${\mathfrak a}^{*}\subseteq {\mathfrak a}$ and claim also ${\mathfrak a}\subseteq {\mathfrak a}^{*}$ . Suppose for the sake of contradiction this is not so. Then let $h\in {\mathfrak a}\setminus {\mathfrak a}^{*}$ 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=\sum _{j}u_{j}a_{j}

of the coefficients of the

f_{j}

. Consider

h_{0}\triangleq \sum _{{j}}u_{{j}}X^{{\deg(h)-\deg(f_{{j}})}}f_{{j}},

which has the same leading term as

h

; moreover

h_{0}\in {\mathfrak a}^{*}

while

h\notin {\mathfrak a}^{*}

. Therefore

h-h_{0}\in {\mathfrak a}\setminus {\mathfrak a}^{*}

and

\deg(h-h'_{0})<\deg(h)

, which contradicts minimality.

Case 2:

\deg(h)=k<d

. Then

a\in {\mathfrak 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

h_{0}\triangleq \sum _{j}u_{j}X^{{\deg(h)-\deg(f_{{j}}^{{(k)}})}}f_{{j}}^{{(k)}},

we yield a similar contradiction as in Case 1.

Thus our claim holds, and ${\mathfrak a}={\mathfrak a}^{*}$ 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.

By induction we see that $R[X_{0},\dotsc ,X_{n-1}]$ will also be Noetherian.
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 ${\mathfrak {a}}\subset R[X_{0},\dotsc ,X_{n-1}]$ 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.
If $A$ is a finitely-generated $R$ -algebra, then we know that $A\simeq R[X_{0},\dotsc ,X_{n-1}]/{\mathfrak {a}}$ , where ${\mathfrak {a}}$ is an ideal. The basis theorem implies that ${\mathfrak {a}}$ must be finitely generated, say ${\mathfrak {a}}=(p_{0},\dotsc ,p_{N-1})$ , 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.

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

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