Nice exercises on Hilbert's basis theorem

Question

I am teaching an extremely basic course in algebraic geometry and I would like to find more exercises on Hilbert's basis theorem, a result from commutative algebra. Unfortunately, I have not been able so far to find interesting exercises in all the books and lecture notes that I've seen.

Could you advise me some exercises on this topic?

Answer 1

A) Obligatory exercise, to be committed to memory.
Any finitely generated algebra over a noetherian ring is noetherian.

B) If $A$ is a PID and $f\in A$ , then the fraction ring $A_f=S^{-1}A$ with $S=\{1,f,f^2,f^3,\cdots\}$ is noetherian.
Amusing example: the ring of all decimal numbers, i.e. those that can be written with finitely many digits after the decimal point (like $-3.1415926535$), is noetherian.

C) The image of $f:\mathbb R\to \mathbb R^n: t\mapsto (t,t^2,\cdots, t^n)$ is an algebraic variety.

D) If $k$ is a field, then any $k$-subalgebra of the $k$-algebra $k[X]$ is noetherian.

E) Is a noetherian algebra over a field finitely generated over that field?

F) a) Is the $\mathbb Q$-algebra $\mathbb Q[1+\sqrt 2,1+2\sqrt 2,\cdots , 1+n\sqrt 2,\cdots ]$ finitely generated over $\mathbb Q$ ?

$\; \; \:$ b) Is the polynomial ring in infinitely many variables $\mathbb Q[X_1,X_2,\cdots , X_n,\cdots ]$ finitely generated over $\mathbb Q$?
$\; \; \:$ c) Is the ring $\mathbb Q(X)$ noetherian? Is it a finitely generated algebra over $\mathbb Q$ ?

NB These are not necessarily corollaries of Hilbert's theorem but are definitely very related to it.

Solution of D)
Let $A\subsetneq k[X]$ be a $k$-subalgebra and let $f(X)=X^n+q_1X^{n-1}+\cdots+q_n\in A \quad (n\geq 1)$.
Then the monic polynomial $T^n+q_1T^{n-1}+\cdots+q_n-f(X)\in (k[f(X)])[T]$ kills $X$, so that $k[X]$ is integral over $k[f(X)]$ and thus finite over $k[f(X)]$ ( Atiyah-Macdonald,Remark page 60).
And now for the killing: $k[f(X)]$ is a noetherian ring by A) and $k[X]$ is a noetherian module over $k[f(X)]$ by finiteness (Atiyah-Macdonald, Proposition 6.5).
Then the $k[f(X)]$-submodule $A$ of $k[X]$ is finitely generated over $k[f(X)]$ as a module and a fortiori as an algebra over the noetherian ring $k[f(X)]$.
Reapplying A), we see that $A$ is a noetherian ring. Et voilà!

Edit: Warning!
The assertion D) becomes false if the field $k$ is replaced by a ring, even a nice noetherian one like $\mathbb Z$:
For example the $\mathbb Z$-subalgebra (which just means the subring!) $A\subset \mathbb Z[X]$ defined by $$A=\mathbb Z\oplus 2\mathbb Z \cdot X\oplus 2\mathbb Z \cdot X^2\oplus 2\mathbb Z \cdot X^3\oplus\cdots =\mathbb Z\oplus (\bigoplus_{i\geq 1} 2\mathbb Z \cdot X^i)$$ is not noetherian since its ideal $$ \langle 2X,2X^2,2X^3,\cdots\rangle \subset A$$ is not finitely generated .

Answer 2

The following results, although easy, are used in "absolute Noetherian reduction" and can be proved by the Hilbert basis theorem:

• Every ring is a filtered colimit of Noetherian rings.

• Every $k$-algebra is a filtered colimit of Noetherian $k$-algebras for a field $k$.

If you have not introduced the language "filtered colimit", you can just say "union of subrings" instead.

You can have them adapt the proof of the Hilbert basis theorem to show the "formal Hilbert basis theorem" that if $A$ is Noetherian, then so is $A[[x]]$.

Answer 3

Exercise: Show that a system of infinitely many polynomial equations can be reduced to a system of finitely many polynomial equations.

Indeed, let $S$ be a set of polynomials. Then $(S)$ is an ideal and hence finitely generated, say with generators $P_1, \dots, P_n$. We have

$$V(S) = V((S)) = V(P_1, \dots, P_n)$$

and we have reduced any system of polynomials to a finite set of polynomials.

Answer 4

Not an application but a variation on the proof technique: show that if $A$ is a noetherian commutative ring, then so is the ring of formal power series $A[[t]]$.

If you are teaching algebraic geometry, then one should mention that every zero set (in $n$-dimensional affine space over a field) is the union of finitely many irreducible zero sets, i.e. zero sets coming from prime ideals. This is essentially the geometric translation of the statement that in a noetherian ring, every radical ideal is the intersection of finitely many prime ideals.

Answer 5

Here's an exercise from Ravi Vakil. The game of Chomp starts with a half-infinite checkboard with squares labeled $(n, m)$ for integers $n, m\geq 0$, with a cookie on each square. Two players take turns choosing a cookie (i.e., not an empty square) and eating all the cookies on that square and the ones that are not to the left or below it; that is, if a player chooses the cookie on $(n, m)$, then they eat all the cookies on $(n', m')$ for $n' \geq n$ and $m'\geq m$. The cookie on $(0, 0)$ is poisoned, and the player who eats it immediately loses. Prove that any game of Chomp ends after finitely many moves.

For the connection with the Hilbert basis theorem, consider $(n, m)$ as representing the monomial $X^n Y^m\in R[X, Y]$. More generally, consider the analogous game on $(\mathbb{Z}^{\geq 0})^d$.