The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.113.1. Let $\mathfrak m$ be a maximal ideal in $k[x_1, \ldots , x_ n]$. The ideal $\mathfrak m$ is generated by $n$ elements. The dimension of $k[x_1, \ldots , x_ n]_{\mathfrak m}$ is $n$. Hence $k[x_1, \ldots , x_ n]_{\mathfrak m}$ is a regular local ring of dimension $n$.

Proof. By the Hilbert Nullstellensatz (Theorem 10.33.1) we know the residue field $\kappa = \kappa (\mathfrak m)$ is a finite extension of $k$. Denote $\alpha _ i \in \kappa $ the image of $x_ i$. Denote $\kappa _ i = k(\alpha _1, \ldots , \alpha _ i) \subset \kappa $, $i = 1, \ldots , n$ and $\kappa _0 = k$. Note that $\kappa _ i = k[\alpha _1, \ldots , \alpha _ i]$ by field theory. Define inductively elements $f_ i \in \mathfrak m \cap k[x_1, \ldots , x_ i]$ as follows: Let $P_ i(T) \in \kappa _{i-1}[T]$ be the monic minimal polynomial of $\alpha _ i $ over $\kappa _{i-1}$. Let $Q_ i(T) \in k[x_1, \ldots , x_{i-1}][T]$ be a monic lift of $P_ i(T)$ (of the same degree). Set $f_ i = Q_ i(x_ i)$. Note that if $d_ i = \deg _ T(P_ i) = \deg _ T(Q_ i) = \deg _{x_ i}(f_ i)$ then $d_1d_2\ldots d_ i = [\kappa _ i : k]$ by Fields, Lemmas 9.7.7 and 9.9.2.

We claim that for all $i = 0, 1, \ldots , n$ there is an isomorphism

\[ \psi _ i : k[x_1, \ldots , x_ i] /(f_1, \ldots , f_ i) \cong \kappa _ i. \]

By construction the composition $k[x_1, \ldots , x_ i] \to k[x_1, \ldots , x_ n] \to \kappa $ is surjective onto $\kappa _ i$ and $f_1, \ldots , f_ i$ are in the kernel. This gives a surjective homomorphism. We prove $\psi _ i$ is injective by induction. It is clear for $i = 0$. Given the statement for $i$ we prove it for $i + 1$. The ring extension $k[x_1, \ldots , x_ i]/(f_1, \ldots , f_ i) \to k[x_1, \ldots , x_{i + 1}]/(f_1, \ldots , f_{i + 1})$ is generated by $1$ element over a field and one irreducible equation. By elementary field theory $k[x_1, \ldots , x_{i + 1}]/(f_1, \ldots , f_{i + 1})$ is a field, and hence $\psi _ i$ is injective.

This implies that $\mathfrak m = (f_1, \ldots , f_ n)$. Moreover, we also conclude that

\[ k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ i) \cong \kappa _ i[x_{i + 1}, \ldots , x_ n]. \]

Hence $(f_1, \ldots , f_ i)$ is a prime ideal. Thus

\[ (0) \subset (f_1) \subset (f_1, f_2) \subset \ldots \subset (f_1, \ldots , f_ n) = \mathfrak m \]

is a chain of primes of length $n$. The lemma follows. $\square$


Comments (2)

Comment #2237 by David Savitt on

Why not use part (2) of [00GB] and induction on n to do this in just a few lines?

Comment #2272 by on

OK, I think I see what you mean, but I am too lazy to change this now.


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