In Atiyah-Macdonald, the authors prove that if $V$ is an irreducible variety over an algebraically closed field $k$, then the local dimension of $V$ (i.e. the Krull dimension of the localization of the coordinate ring at any point) is equal to the transcendence degree of the function field of $V$ over $k$ (Theorem 11.25). Then, in Exercise 3 of Chapter 11 the authors ask the reader to extend this result to non-algebraically-closed fields. I don't see why algebraically-closed is necessary for their proof, so I want to ask why the following does not work:
Proof: Let $k[x_1, \dots, x_n]$ be a polynomial ring in $n$ variables over any field $k$. Let $\mathfrak{p}$ be a prime of $k[x_1, \dots, x_n]$, and $A(V) = k[x_1, \dots, x_n] / \mathfrak{p}$. Let $k(V)$ denote the field of fractions of $A(V)$. Let $P = (a_1, \dots, a_n)$ be a point of the variety $V$ defined by $\mathfrak{p}$. Then the maximal ideal $(x_1 – a_1, \dots, x_n – a_n)$ of $k[x_1, \dots, x_n]$ descends to a maximal ideal $\mathfrak{m}$ of $A(V)$. By Noether's normalization lemma, we can find $y_1, \dots, y_d$ algebraically independent elements in $A(V)$ such that $A(V)$ is integral over $B = k[y_1, \dots, y_d]$. Thus $\mathfrak{n} = \mathfrak{m} \cap B$ is a maximal ideal of $B$. Further, $B$ is a UFD since a polynomial ring over any field is a UFD, so $B$ is integrally closed. Thus Atiyah Macdonald Lemma 11.26 shows that $\dim A_\mathfrak{m} = \dim B_\mathfrak{n}$, but $\dim B_\mathfrak{n} = d$ by the example on page 124 of Atiyah Macdonald. QED.
This is essentially the exact proof of Theorem 11.25 in A&M – it seems to me they are only using that $k$ is algebraically closed for the weak Nullstellensatz, but even when $k$ is not algebraically closed, the points of $V$ still correspond to maximal ideals. The correspondence just might not be surjective. Is there some other reason we need $k$ to be algebraically closed that I am missing in the above proof?
Best Answer
I think the problem in your proof does, in the end, lie in the fact that not every maximal ideal of $k[x_1, \ldots, x_n]$ is necessarily of the form $(x_1 - a_1, \ldots, x_n - a_n)$ for a non-algebraically-closed field $k$. The last step of your proof, where you claim that $\dim{B_\mathfrak{n}} = d$, is not actually demonstrated anywhere in Atiyah-MacDonald for an arbitrary maximal ideal $\mathfrak{n}$. What is demonstrated (page 121) is that the ring $k[x_1, \ldots, x_n]$ localized at a maximal ideal of the form $(x_1 - a_1, \ldots, x_n - a_n)$ is of dimension $n$, which implies, of course, that $\dim{k[x_1, \ldots, x_n]_{\mathfrak{m}}} = n$ for all maximal ideals $\mathfrak{m}$ when $k$ is algebrically closed. (Furthermore, the example you cite on page 124 has nothing to do with the dimension of $B_\mathfrak{n}$; rather, it stipulates that $B_{(x_1, \ldots, x_n)}$ is regular.)
To modify your proof as it is, you would either have to show that every maximal ideal in $B$ is of the form $(x_1 - a_1, \ldots, x_n - a_n)$ (which is untrue), or somehow show that $\dim{B_\mathfrak{n}} = d$ for an arbitrary maximal ideal $\mathfrak{n} \subseteq k[x_1, \ldots, x_d]$ independently.
Edit: Perhaps the third sentence of your proof ("Then the maximal ideal $(x_1−a_1,\ldots,x_n−a_n)$ of $k[x_1,\ldots,x_n]$ descends to a maximal ideal $\mathfrak{m}$ of $A(V)$") was an attempt to remedy this error; if it was, I couldn't decipher exactly what you meant.