Dickson's Lemma (proof of Prop. 2.23 in Hasset's Intro to Alg Geom)

Question

I'm studying Hasset's book by myself but I had no previous formal algebra training. To prove Dickson's lemma (prop. 2.23, p. 19) he defines the auxiliary monomial ideals $$J_m=\left<x^\alpha \in k[x_1,\ldots,x_n]: x^\alpha y^m \in J\right>,$$ where $J$ is a monomial ideal. On the next page (on the completion of the proof) he writes that the sequence of $J_m$ terminates. It's probably something obvious that I'm missing but I'm having a hard time seeing why this is the case.

Answer 1

To prove the indicated statement from scratch, we first sketch a proof of another result that's called Dickson's lemma: For any infinite sequence of distinct $\alpha_0, \alpha_1\in \mathbb{N}^n$, there exist $i < j$ such that $\alpha_i \leq \alpha_j$ (that is, each component $\alpha_i^k \leq \alpha_j^k$.) Assume the result holds for $(n-1)$-tuples. Consider the set of last coordinates of the $\alpha_i$. If this set is bounded, then we can choose some infinite subset of the $\alpha_i$ with constant last coordinate and apply the result for $n - 1$. Otherwise, we can choose an infinite sequence of the $\alpha_i$ for which the last coordinate is strictly increasing. Now we can again apply the result for $n - 1$. The lemma follows by induction on $n$.

To prove the original statement, suppose instead that the chain $J_0\subset J_1\subset \dots$ doesn't stabilize. Assume without loss of generality (by relabelling if necessary) that the $J_i$ are distinct. The $J_i$ are ideals, so if $\alpha \leq \beta$ and $x^\alpha\in J_i$, then $x^\beta\in J_i$ as well. Choose some $\alpha_i\in J_i$ such that $\alpha_i\not\in J_k$ for $k < i$. Then $\alpha_i\not\leq \alpha_j$ for distinct $i, j$, contradicting Dickson's lemma.

Answer 2

The ideals $J_0 \subset J_1 \subset J_2 \subset \cdots$, and $k[x_1, \dots, x_n]$ is Noetherian.