Saturday, May 2, 2015

Model theory and algebraic geometry, 2 — Definable sets, types; quantifier elimination

This is the second post in a series of 4 devoted to the exposition of interactions between model theory and algebraic geometry. In the first one, I explained the notions of language, structures and theories, with examples taken from algebra. Here, I shall discuss the notion of definable set, of types, as well as basic results from dimension theory ($\omega$-stability).

So we fix a theory $T$ in a language $L$. A definable set is defined, in a given model $M$ of $T$, by a formula. More precisely, we consider definable sets in cartesian powers $M^n$ of the model $M$, which can be defined by a formula in $n$ free variables with parameters in some subset $A$ of $M$. By definition, such a formula is a formula of the form $\phi(x;a)$, where $\phi(x;y)$ is a formula in $n+m$ free variables, split into two groups $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_m)$ and $a=(a_1,\dots,a_m)\in A^m$ is an $m$-tuple of parameters; the formula $\phi(x;y)$ can have quantifiers and bounded variables too. Given such a formula, we define a subset $[\phi(x;a)]$ of $M^n$ by $\{ x\in M^n\mid \phi(x;a)\}$. We write $\mathrm{Def}(M^n;A)$ for the set of all subsets of $M^n$ which are definable with parameters in $A$.

Let us give examples, where $L$ is the language of rings and $T$ is the theory $\mathrm{ACF}$ of algebraically closed fields:
  • $V_1=\{x\mid x\neq 0 \}\subset M $, given by the formula “$x\neq 0$” with 1 variable and $0$ parameter;
  • $V_2=\{x\mid \exists y, 2xy=1\} \subset M $, given by the formula “$\exists y, 2xy=1$” with 1 free variable $x$, and one bounded variable $y$;
  • $V_3=\{(x,y)\mid x^2+\sqrt 2 y^2=\pi \}\subset \mathbf C^2$, where the model $\mathbf C$ is the field of complex numbers, $\phi((x,y),(a,b))$ is the formula $x^2+ay^2=b$ in 4 free variables, and the parameters are given by $(a,b)=(\sqrt 2,\pi)$.
Theorem (Chevalley). — Let $L$ be the language of rings, $T=\mathrm{ACF}$ and $M$ be an algebraically closed field; let $A$ be a subset of $M$. The set $\mathrm{Def}(M^n;A)$ is the smallest boolean algebra of subsets of $M^n$ which contains all subsets of $M^n$ of the form $[P(x;a)]$ where $P$ is a polynomial in $n+m$ variables with coefficients in $\mathbf Z$ and $a=(a_1,\dots,a_m)$ is an $m$-tuple of elements of $A$. In other words, a subsets of $M^n$ is definable with parameters in $A$ if and only if it is constructible with parameters in $A$.

The reason behind this theorem is the following set-theoretic interpretation of quantifiers and logical connectors. Precisely, if $\phi$ is a formula in $n+m+p$ variables, and $a\in A^p$, the definable subset $[\exists y \phi(x,y,a)]$ of $M^n$ coincides with the image of the definable subset $[\phi(x,y;a)]$ of $M^{n+m}$ under the projection $p_x \colon M^{n+m}\to M^n$. Similarly, if $\phi(x)$ and $\psi(x)$ are two formulas in $n$ free variables, then the definable subset $[\phi(x)\wedge\psi(x)]$ is the union of the definable subsets $[\phi(x)]$ and $[\psi(x)]$. And if $\phi(x)$ is a formula in $n$ variables, then the definable subset $[\neg\phi(x)]$ is the complement in $M^n$ of the definable subset $[\phi(x)]$.

For example, the subset $V_2=[\exists y, 2xy=1]$ defined above can also be defined by $M\setminus [2x=0]$.

One says that the theory ACF admits elimination of quantifiers: modulo the axioms of algebraically closed fields, every formula of the language $L$ is equivalent to a formula without quantifiers.

An important consequence of this property is that for every extension $M\hookrightarrow M'$ of models of ACF, the theory of $M'$ is equal to the theory of $M$—one says that every extension of models is elementary.

Let $p$ be either $0$ or a prime number. Observe that every algebraically closed field of characteristic $p$ is an extension of $\overline{\mathbf Q}$ if $p=0$, or of $\overline{\mathbf F_p}$ if $p$ is a prime number. As a consequence, for every characteristic $p\geq0$, the theory $\mathrm{ACF}_p$ of algebraically closed fields of characteristic $p$ (defined by the axioms of $\mathrm{ACF}$, and  the axiom $1+1+\dots+1=0$ that the characteristic is $p$ if $p$ is a prime number, or the infinite list of axioms that assert that the characteristic is $\neq \ell$, if $p=0$) is complete: this list of axioms determines everything that can be said about algebraically closed fields of characteristic $p$.

Definition. — Let $a\in M^n$ and let $A$ be a subset of $M$. The type of $a$ (with parameters in $A$) is the set $\mathrm{tp}(a/A)$ of all formulas $\phi(x;b)$ in $n$ free variables with parameters in $A$ such that $\phi(a;b)$ holds in the model $M$.

Definition. — Let $A$ be a subset of $M$. For every integer $n\geq 0$, the set $S_n(A)$ of types (with parameters in $A$) is the set of all types $\mathrm{tp}(a/A)$, where $N$ is an extension of $M$ which is a model of $T$ and $a\in N^n$. One then says that this type is realized in $N$.

Gödel's completeness theorem allows us to give an alternative description of $S_n(A)$. Namely, let $p$ be a set of formulas in $n$ free variables and parameters in $A$ which contains the diagram of $A$ (that is, all formulas which involve only elements of $A$ and are true in $M$). Assume that $p$ is consistent (there exists a model $N$ which is an extension of $M$ and and element $a\in M^n$ such that $\phi(a)$ holds in $N$ for every $\phi\in p$) and maximal (for every formula $\phi\not\in p$, then for every model $N$ and every $a\in N^n$ such that $p\subset \mathrm{tp}(a/A)$, then $\phi(a)$ does not hold). Then $p\in S_n(A)$.

For every formula $\phi\in L(A)$ in $n$ free variables and parameters in $A$, let $V_\phi$ be the set of types $p\in S_n(A)$ such that $\phi\in p$. Then the subsets $V_\phi$ of $S_n(A)$ consistute a basis of open sets for a natural topology on $S_n(A)$.

Theorem. — The topological space $S_n(A)$ is compact and totally discontinuous.

Let us detail the case of the theory ACF in the langage of rings. I claim that if $K$ is a field, then $S_n(K)$ is homeomorphic to the spectrum $\mathop{\rm Spec}(K[T_1,\dots,T_n])$ endowed with its constructible topology. Concretely, for every algebraically closed extension $M$ of $K$ and every $a\in M^n$, the homeomorphism $j$ maps $\mathrm{tp}(a/K)$ to the prime ideal $\mathfrak p_a$ consisting of all polynomials $P\in K[T_1,\dots,T_n]$ such that $P(a)=0$.

A type $p=\mathrm{tp}(a/K)$ is isolated if and only if the prime ideal $\mathfrak p_a$ is maximal. Consequently, if $n=1$, there is exactly one non-isolated type in $S_1(K)$, corresponding to the generic point of the spectrum $\mathop{\rm Spec}(K[T])$.

As for any compact topological space, a space of types can be studied via its Cantor-Bendixson analysis, which is a decreasing sequence of subspaces, indexed by ordinals, defined by transfinite induction. First of all, for every topological space $X$, one denotes by $D(X)$ the set of all non-isolated points of $X$. One then defines $X_0=X$, $X_{\alpha}=D(X_\beta)$ if $\alpha=\beta+1$ is a successor-ordinal, and $X_\alpha=\bigcap_{\beta<\alpha} X_\beta$ if $\alpha$ is a limit-ordinal. For $x\in X$, the Cantor-Bendixson rank of $x$ is defined by $r_{CB}(x)=\alpha$ if $x\in X_\alpha$ and $x\not\in X_\beta$ for $\beta>\alpha$, and $r_{CB}(x)=\infty$ if $x\in X_\alpha$ for every ordinal $\alpha$. The set of points of infinite rank is the largest perfect subset of $X$.

Let us return to the example of the theory ACF. If a type $p\in S_n(K)$ corresponds to a prime ideal $\mathfrak p=j(p)$ of $\mathop{\rm Spec}(K[T_1,\dots,T_n])$, its Cantor-Bendixson rank is the Zariski dimension of $V(I)$. More generally, if $F$ is a constructible subset of $\mathop{\rm Spec}(K[T_1,\dots,T_n])$, then $r_{CB}(F)$ is the Zariski-dimension of the Zariski-closure of $F$. Moreover, the points of maximal Cantor-Bendixson rank correspond to the generic points of the irreducible components of maximal dimension; in particular, there are only finitely many of them.

Definition. — One says that a theory $T$ is $\omega$-stable if for every finite or countable set of parameters $A$, the space of 1-types $S_1(A)$ is finite or countable.

The theory ACF is $\omega$-stable. Indeed, if $K$ is the field generated by $A$, then $K[T]$ being
a countable noetherian ring, it has only countably many prime ideals.

Since any non-empty perfect set is uncountable, one has the following lemma.

Lemma. — Let $T$ be an $\omega$-stable theory and let $M$ be a model of $T$. Then the Cantor-Bendixson rank of every type $x\in S_n(M)$ is finite.

Let us assume that $T$ is $\omega$-stable and let $F$ be a closed subset of $S_n(M)$. Then $r_{CB}(F)=\sup \{ r_{CB}(x)\,;\, x\in F\}$ is finite, and the set of points $x\in F$ such that $r_{CB}(x)=r_{CB}(F)$ is finite and non-empty.

This example gives a strong indication that the model theory approach may be extremly fruitful for the study of algebraic theories whose geometry is not as well developed than algebraic geometry.

2 comments :

  1. There seems to be a typo in the statement of Chevalley's theorem: "smallest boolean _algebra_ ..."

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