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I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem.

http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/

This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb C^n$ is bijective. It seems to me that the only point in the proof presented in the post that is not explained completely is the following lemma:

Lemma. Take any finitely generated ring over $\mathbb Z$ and quotient it by a maximal ideal. Then the quotient is a finite filed.

Is there some comprehensible reference for the proof of this lemma?

In slightly different wording, the question is the following: assuming Nullstelensatz, can one really give a complete proof of Ax-Grothendick theorem in two pages, so that it can be completely explained in one (2 hours) lecture of an undergraduate course on algebraic geometry?

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This was previously answered: mathoverflow.net/questions/30599/… –  fherzig Mar 14 '11 at 22:19
    
I still wonder where this little Lemma appears in the literature ... –  Martin Brandenburg Oct 13 '13 at 8:52
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4 Answers 4

up vote 12 down vote accepted

To prove Nullstellensatz over $\mathbb{Z}$: as the morphism $f: \mathrm{Spec}(R)\to\mathrm{Spec}(\mathbb Z)$ is of finite type, a theorem of Chevalley says that the image of any constructible subset is constructible. So the image of any closed point by $f$ is a point which is a constructible subset. This can not be the generic point of $\mathrm{Spec}(\mathbb Z)$, so it must be a closed point.

Note that this does not hold in general. For example, over the ring of $p$-adic integers, the ideal $(pX-1)\mathbb{Z}_p[X]$ is maximal, but its preimage in $\mathbb{Z}_p$ is $0$ and it not maximal.

[EDIT] Another proof using Noether's normalization lemma: Noether's normalization lemma over a ring A: if a maximal ideal $\mathfrak m$ of $R$ is such that $\mathfrak m\cap \mathbb Z=0$, then $R/\mathfrak m$ is finite type over (and contains) $\mathbb Z$. So there exits $f\in\mathbb Z$ non-zero and a finite injective homomorphism $\mathbb Z_f[X_1,\dots, X_d]\hookrightarrow R/\mathfrak m$. But then $\mathbb Z_f[X_1,\dots, X_d]$ must be a field. This is impossible because the units of this ring are $\pm f^k$, $k$ relative integers.

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This first argument is quite nice! –  Daniel Litt Mar 6 '11 at 2:49
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The theorem of Chevalley can be proved by induction, using the second argument :). –  Qing Liu Mar 6 '11 at 10:17
    
@Qing Liu, thanks for the answer. I am lost when you say in lines 2-3 of Edit: "So there exits $f\in \mathbb Z$ non-zero and a finite injective homomorphism ...". Why is there such $f$? –  aglearner Mar 7 '11 at 0:16
    
Sorry, my punctions were not good. I should write "If a maximal ideal ..., so there exists f". The existence of such a $f$ is a form of Noether's normalization lemma that you can find at mathoverflow.net/questions/42276 –  Qing Liu Mar 7 '11 at 8:14
    
After one year and half I guess I understand the logic of both answers. Could you please say if there is a pedagogic explanation of Chevalet's theorem somewhere on the web? –  aglearner Oct 14 '12 at 13:34
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Let $R$ be a finitely generated $\mathbb{Z}$-algebra, and $\mathfrak{m}\subset R$ are maximal ideal. We wish to show $R/\mathfrak{m}$ is a finite field.

Let $i: \mathbb{Z}\to R$ be the unique ring map; then $i^{-1}(\mathfrak{m})$ is a maximal ideal in $\mathbb{Z}$ (as $R$ is finitely generated over $\mathbb{Z})$, and thus $\mathbb{Z}/i^{-1}(\mathfrak{m})$ is a finite field $\mathbb{F}_p$ for some prime $p$. As $R$ is finitely generated over $\mathbb{Z}$, $R/\mathfrak{m}$ is finitely generated over $\mathbb{F}_p$. But all finite field extensions of $\mathbb{F}_p$ are still finite, completing the proof.

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Daniel, thanks for the proof of the lemma! –  aglearner Mar 6 '11 at 0:16
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Perhaps it's worth emphasizing that what makes this argument work is the non-trivial fact (proved in Qing Liu's answer) that if $f:\mathbb{Z}\rightarrow\mathbb{R}$ is of finite type, then the pre-image of a maximal ideal of $R$ under $f$ is non-zero. This is a particular case of the general Nullstellensatz on Jacobson rings (found in, e.g., Eisenbud's book). A proof of the statement about fields finitely generated as rings using the usual Nullstellensatz is outlined in Exercise 6 of the Noetherian rings chapter of Atiyah and Macdonald. –  Keenan Kidwell Mar 6 '11 at 1:07
    
@Keenan Kidwell: Of course; thanks for making that explicit. –  Daniel Litt Mar 6 '11 at 1:13
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An ignorant question: "$R/\mathfrak{m}$ is finitely generated over $\mathbb F_p$. But all finite field extensions of $\mathbb F_p$ are still finite, completing the proof". I'm missing something here. How did finitely generated extension get to be the same as finite field extension? (in particular isn't $\mathbb F_p(x)$ a finitely generated extension but not a finite extension)? –  Anthony Quas Mar 6 '11 at 21:44
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$\mathbb{F}_p(x)$ is not finitely generated as an $\mathbb{F}_p$-algebra; in particular, all the irreducibles need to be inverted. See e.g. mathreference.com/ag,fgaf.html –  Daniel Litt Mar 6 '11 at 21:54
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One can give a more elementary proof of the fact that $\mathfrak{m} \cap \mathbb{Z} \neq 0$ - By more elementary I mean a proof that only uses the Nullstellensatz over $\mathbb{Q}$.

Notice that it is enough to verify the claim for $R=\mathbb{Z}[x_1,..,x_n]$, and $\mathfrak{m} \in Max(R)$.

Suppose there is $\mathfrak{m} \in Max(R)$ such that $\mathfrak{m} \cap \mathbb{Z} =0$. Then, we may assume that $\mathbb{Z} \subseteq F :=\mathbb{Z}[x_1,..,x_n]/\mathfrak{m}$. If we denote by $\alpha_{i}=x_i+\mathfrak{m}$ we have that $F=\mathbb{Z}[\alpha_1,..,\alpha_n]$. Since $F$ is a field we conclude that $\mathbb{Z}[\alpha_1,..,\alpha_n]=\mathbb{Q}(\alpha_1,..,\alpha_n)$.

Claim: $F/\mathbb{Q}$ is an algebraic extension.

proof: $F/\mathbb{Q}$ is a finitely generated field extension- generated as an algebra- in particular $F$ is of the form $\mathbb{Q}[y_1,..,y_m]/M$ for some $M$ maximal ideal of $\mathbb{Q}[y_1,..,y_m].$ By the Nullstellensatz $M$ has a zero $(\beta_1,...,\beta_m)$ where each $b_i$ is algebraic over $\mathbb{Q}$, so $F=\mathbb{Q}(\beta_1,...,\beta_m)$ is algebraic over $\mathbb{Q}$.

Since each $\alpha_{i}$ is algebraic, there are integers $q_i$'s such that $q_{i}\alpha_{i}$ is integral over $\mathbb{Z}$ for all $i$. In particular $F=\mathbb{Z}[\alpha_1,..,\alpha_n]$ is an integral extension of $\displaystyle \mathbb{Z}[\frac{1}{q_1},..,\frac{1}{q_n}]$. Since $F$ is a field we have that $\displaystyle \mathbb{Z}[\frac{1}{q_1},..,\frac{1}{q_n}]$ is a field, which is a contradiction( $p$ is not invertible for any prime not dividing $q_{1}...q_{n}$).

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Unfortunately, I can not understand when you write " By the Nullstellensatz we have that each $\alpha_i$ is algebraic over $\mathbb Q$". Could you please explain this point? Are you using Nullstelensatz over $\mathbb Q$ here? To which ring are you applying it? –  aglearner Mar 7 '11 at 0:12
    
@aglearner: I've added an explanation to what you are wondering. The point is that one version of the Nullstellensatz, which I learned by the name algebraic Nullstellensatz, is the following: A finitely generated extensions of fields $F/K$ is algebraic. –  Guillermo Mantilla Mar 7 '11 at 1:21
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This is not an answer to your question, but let me point out that the Ax-Grothendieck theorem is now easy to prove using E-polynomials (Hodge-Deligne polynomials). If $f:X \to X $ is an injective endomorphism of a complex algebraic variety, then $E(X) = E(f(X))=E(X)-E(X\setminus f(X))$. So $E(X\setminus f(X))=0$ and $X\setminus f(X) = \emptyset$, because the degree of a constructible set is twice its dimension. Since one supposes the mixed Hodge theory, this proof is not trivial at all. But at least for me, this looks more natural.

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I hadn't heard about this, and I don't know anything about E-polynomials. Do you know if the proof you describe uses anything in characteristic p in the background? Any "spreading" over Z? –  Marty Mar 6 '11 at 3:17
    
This proof uses only the mixed Hodge theory, and one does not need to switch the base field or ring. On the other hand, both proofs by E-polynomial and finite fields have the "motivic" nature. Namely the E-polynomial and the number of rational points are generalized Euler charqcteristics, that is, they have the additivity and multipicativity. –  Takehiko Yasuda Mar 6 '11 at 9:39
    
But I wonder, since much of mixed Hodge theory (after Deligne) lifts characteristic p results to get characteristic zero results. And perhaps even transcendental proofs in Hodge theory (like those of Saito) might hide some characteristic p aspects. Do you know if characteristic p is hidden in the background for the results on the E-polynomial? –  Marty Mar 10 '11 at 22:05
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