Robinson’s characteristic p principle and the Ax-Grothendieck theorem

Model theory often provides a framework from one which one can obtain “finitary” versions of infinitary results, and vice versa.

One spectacular example is the Ax-Grothendieck theorem, which states that an injective polynomial map ${P: \mathbb{C}^n \rightarrow \mathbb{C}^n}$ is surjective. The key idea here is that the theorem for polynomial maps of a fixed degree is a statement of first-order logic, to which the compactness theorem applies. Next, the theorem is trivial when ${\mathbb{C}}$ is replaced by a finite field, and one then deduces it for ${\overline{\mathbb{F}_p}}$ (and maps ${P: \overline{\mathbb{F}_p} \rightarrow \overline{\mathbb{F}_p}}$ by an inductive limit argument. It then holds for algebraically closed fields of nonzero characteristic, because ${ACF_p}$ is a complete theory—any first-order statement true in one algebraically closed field of characteristic ${p}$ is true in any such field. Finally, one appeals to a famous result of Abraham Robinson that any first-order statement true in algebraically closed fields of characteristic ${p>p_0}$ is true in algebraically closed fields of characteristic zero.

There is a discussion of this result and other proofs by Terence Tao here.

For fun, I will formally state and prove Robinson’s theorem.

Theorem 1 (A. Robinson) Let ${S}$ be a statement in first-order logic in the language of fields (i.e., referring to the operations of addition and multiplication, and the constants ${0,1}$ ). Then ${S}$ is true in algebraically closed fields of characteristic zero if and only if ${S}$ is true algebraically closed fields of arbitrarily high (or sufficiently high, ${p>p_0}$ ) characteristic ${p}$ .

One can give a direct result of one half of this theorem using ultraproducts and Los’s theorem, but we can also appeal to the compactness theorem. Let ${A}$ be the set of axioms of an algebraically closed field. For each prime ${p}$ , let ${R_p}$ be the sentence ${1 + 1 + \dots + 1 = 0}$ ( ${p}$ times).

Suppose ${S}$ is true in an algebraically closed field of characteristic zero and hence in all; by Vaught’s test, the theory of algebraically closed fields of a fixed characteristic is complete. Then ${A \cup \{ \neg R_p \} }$ implies ${S}$ ; a finite subset of these axioms thus must imply ${S}$ (this is a consequence of the compactness theorem). In particular, ${S}$ is implied by ${A}$ together with some finite number of the conditions ${\{ \neg R_p\}}$ , which means it is true in algebraically closed fields of characteristics outside a finite number of primes.

Conversely, suppose ${S}$ is true in algebraically closed fields of arbitrarily high characteristic. Then any finite subset of ${A \cup \{ \neg R_p \} \cup S}$ is satisfiable, so by compactness, the whole collection is satisfiable. Therefore, ${S}$ holds in some algebraically closed field of characteristic zero.

Here is another application:

Corollary 2 Let ${P(X_1, \dots, X_n ) \in \mathbb{Z}[X_1, \dots, X_n]}$ be absolutely irreducible, i.e. irreducible in ${\mathbb{C}[X_1, \dots, X_n]}$ . Then ${\bar{P}}$ is absolutely irreducible in ${ \mathbb{F}_p[X_1, \dots, X_n]}$ for almost all primes ${p}$ .

Indeed, the statement about irreducibility can be made into a first-order property. Note that absolute irreducibility is necessary! Consider ${X^2-2}$ ; this is irreducible in ${\mathbb{Q}[X]}$ , but it is reducible in ${\mathbb{F}_p[X]}$ if ${2}$ is a quadratic residue modulo ${p}$ . In fact, the theory of fields of a specified characteristic is far from complete.

Finally, I will formally prove a stronger variant of the Ax-Grothendieck theorem:

Theorem 3 Let ${V}$ be an algebraic set defined over an algebraically closed field ${F}$ , let ${P: V \rightarrow V}$ be a morphism. Suppose it is injective. Then it is surjective.

Consider the statement ${S_{m,d,n}}$ : “Whenever ${Q_1, \dots, Q_m, P_1, \dots, P_n}$ are polynomials of degree at most ${d}$ in ${n}$ -variables satisfying the following condition: for all ${x_1, \dots, x_n, y_1, \dots, y_n \in F}$ with ${Q_1(x)=Q_2(x)=\dots=Q_m(x)=Q_1(y)=\dots=Q_m(y)=0}$ and some ${x_i \neq y_i}$ , then some ${P_j(x) \neq P_j(y)}$ , we have the following conclusion: for any ${z_1, \dots, z_n}$ satisfying each ${Q_j}$ , there exist ${w_1, \dots, w_n}$ satisfying each ${Q_j}$ such that ${P}$ maps ${w}$ to ${z}$ .” This is clearly a first-order condition, and it holds in each finite field because an injective map on a finite set is surjective! It thus holds in the algebraic closure of each finite field by a direct limit argument. It thus holds in algebraically closed fields of characteristic zero by Robinson’s theorem. Taking the conjunction of all ${S_{m,d,n}}$ , one obtains the theorem.

Source: David Marker, Model Theory: An Introduction

. Says:

July 19, 2010 at 1:44 am

I’ve always found the model-theoretic proofs of these things to be a red herring, and quite irrelevant for the practice of this kind of algebraic geometry (not to say model theory doesn’t have interesting applications elsewhere, but for the above kind of stuff it feels like the wrong technique for the task at hand). Grothendieck gives very elegant purely algebraic and rather “concrete” proofs in EGA IV (no use of any logic, ultraproducts, etc.), in fact (more importantly) of *vast* generalizations which are very useful.

For example, if $f:X \rightarrow S$ is a finitely presented morphism of schemes with $S$ integral and if the generic fiber is geometrically irreducible (resp. geometrically reduced, resp. geometrically connected, resp. lots of other useful properties of *geometric* fibers) then the same holds for all fibers over a dense open in the base. Can one actually squeeze the case of geometric reducedness of fibers (let alone the zillion other properties EGA considers, like geometric normality, or even geometric irreducibility when not using hypersurfaces) out of Robinson-type stuff? Likewise, if $h:X \rightarrow X$ is an $S$-morphism which is radiciel (resp. a monomorphism) then it is surjective (resp. an isomorphism); see IV_3 10.4.11 resp. IV_4 17.9.6 in EGA; the radiciel case is a huge generalization of the case of endomorphisms injective on rational points for a variety over an alg. closed field.

Akhil Mathew Says:

July 19, 2010 at 8:28 am
Hm, interesting – thanks for pointing it out. I didn’t know that there was a more general approach to this business, but the fact that Grothendieck’s name is in the theorem makes it rather unsurprising.

Yes, Marker’s book explains how Chevalley’s theorem translates to quantifier elimination for the theory of algebraically closed fields (perhaps of a fixed characteristic; my memory’s hazy), and I’d imagine that implies constructibility of things like geometric irreducibility, but going from there to the full scheme-theoretic result might take extra work, and it does seem more natural to avoid logic (especially when it depends on the axiom of choice). The model-theoretic proof of the Nullstellensatz has always felt a bit strange to me, especially since it is such a concrete statement about polynomials (and I don’t know how model theory leads to the general statement about Jacobson rings).

blah Says:

September 10, 2010 at 3:12 pm
Often times the real use for model theory comes when considering a slightly different notion of “variety”. For instance, when studying difference or differential equations. I would not argue that establishing all of the model theoretic results in these areas by other means is impossible, but I would point out that work of model theorists in these areas lead to many new results and applications in other fields (like diophantine geometry). Also, it is fairly unfair to expect model theory to prove scheme theoretic statements, since some of these are inherently not first order.

Climbing Mount Bourbaki