nLab
geometric stability theory

Contents

Idea

Geometric stability theory is the principal part of the branch of model theory called geometric model theory?. It was introduced in works of Boris Zilber, Gregory Cherlin, Ehud Hrushovski, Anand Pillay, and others.

Geometric stability theory has largely to do with the model-theoretic classification of structures in terms of dimension-like quantities that can be axiomatized in terms of notions of combinatorial geometry such as matroid. Key guiding examples include vector spaces and algebraically closed fields, and many of the guiding concepts have an algebro-geometric flavor (e.g., Morley rank as a generalization of Krull dimension).

Such a structural geometric approach can make geometric stability theory an attractive key of entry into modern model theory for those mathematicians who are not already logicians.

Basic concepts

Perhaps the key axiomatic notions of geometric stability theory (which at the outset don’t seem particularly wedded to logic) are those of pregeometry and geometry.

Definition

Let XX be a set. A pregeometry on XX is a closure operator (i.e., a monad cl:PXPXcl \colon P X \to P X on the power set), satisfying the following two conditions:

  • The monad clcl is finitary, i.e., AXA \in X and acl(A)a \in cl(A), then there is a finite A 0AA_0 \subseteq A such that acl(A 0)a \in cl(A_0).

  • (Exchange condition) If APXA \in P X, a,bXa,b \in X, and acl(A{b})a \in cl(A\cup\{b\}), then acl(A)a \in cl(A) or bcl(A{a})b \in cl(A \cup \{\a\}). (Cf. matroid)

A geometry is a pregeometry such that cl()=cl(\emptyset) = \emptyset and cl({x})={x}cl(\{x\}) = \{x\} for all xXx \in X.

Examples
  • Let XX be a vector space, and let clcl be the monad on PXP X whose algebras are vector subspaces of XX. Clearly clcl is finitary (any subspace is the set-theoretic union of finite-dimensional subspaces), and the exchange condition is a classical fact about vector spaces related to the notion of independence. Thus clcl is a pregeometry.

  • Similarly, let XX be a projective space V\mathbb{P}V, and let clcl be the monad on PXP X whose algebras are projective subspaces. Then clcl is a geometry (the closure of a point is a point). Any pregeometry clcl gives rise to a geometry in a similar way, in the sense that a pregeometry clcl induces a geometry on the image of the function XPXX \to P X, xcl({x})x \mapsto cl(\{x\}), as explained in Remark 2.

  • Let XX be an algebraically closed field; let clcl be the monad on PXP X whose algebras are algebraically closed subfields. Then clcl is a pregeometry. That the exchange condition is satisfied is a classical result credited to Steinitz1.

Definition

Given a pregeometry (X,cl)(X, cl), a subset APXA \in P X is independent if for all aAa \in A, acl(A{a})a \notin cl(A - \{a\}). An independent set AA said to be a basis for YPXY \in P X if Ycl(A)Y \subseteq cl(A). All bases of YY have the same cardinality, called the dimension of YY.

Remark

Given a pregeometry (X,cl)(X, cl) and a subset YXY \subseteq X, there is a restriction pregeometry cl Y:P(Y)P(Y)cl^Y: P(Y) \to P(Y) defined by the formula

cl Y(S)=cl(S)Y.cl^Y(S) = cl(S) \cap Y.

It is immediate that AYA \subseteq Y is independent in (X,cl)(X, cl) iff it is independent in (Y,cl Y)(Y, cl^Y), and that it is a basis for YY in (X,cl)(X, cl) iff it is a basis for YY in (Y,cl Y)(Y, cl^Y). By this observation, to prove that any YP(X)Y \in P(X) has a well-defined dimension, we may assume without loss of generality that Y=XY = X. That a pregeometry XX has well-defined dimension was proven here.

Remark

There is a standard way of getting a geometry from a pregeometry (X,cl)(X, cl). First, if cl()cl(\emptyset) \neq \emptyset, then replace XX by XXcl()X' \coloneqq X - cl(\emptyset), equipped with the restriction pregeometry of the previous remark. Then define an equivalence relation on XX' by xyx \sim y if cl({x})=cl({y})cl(\{x\}) = cl(\{y\}), and define a pregeometry on the quotient set X/X'/\sim by

cl^(A){[b]:bcl(A)}\widehat{cl}(A) \coloneqq \{[b]: b \in cl(A)\}

where [b][b] denotes the equivalence class of bb. This abstracts the process of taking a projectivization of a vector space.

Minimal and strongly minimal sets

We’ll suppose for now that L\mathbf{L} is a countable signature, so that the language it generates consists of countably many formulas. Let M\mathbf{M} be an L\mathbf{L}-structure, with underlying set MM.

Definition

For AMA \subseteq M, an element bMb \in M is algebraic over AA if there is a formula ϕ(y,w 1,,w n)\phi(y, w_1, \ldots, w_n) and elements a 1,,a nAa_1, \ldots, a_n \in A such that Mϕ(b,a 1,,a n)\mathbf{M} \models \phi(b, a_1, \ldots, a_n) and

{cM:Mϕ(c,a 1,,a n)}\{c \in M: \mathbf{M} \models \phi(c, a_1, \ldots, a_n)\}

is finite. The algebraic closure of AA, denoted acl(A)acl(A), is the set of elements of MM that are algebraic over AA.

Proposition

The algebraic closure Aacl(A)A \mapsto acl(A) defines a finitary closure operator on MM.

Proof

That aclacl is monotone (preserves order) is obvious. Also the fact that aclacl is finitary is easy: given that bacl(A)b \in acl(A) satisfies Mϕ(b,a 1,,a n)\mathbf{M} \models \phi(b, a_1, \ldots, a_n) for a 1,,a nAa_1, \ldots, a_n \in A, then similarly bacl(A 0)b \in acl(A_0) for A 0={a 1,,a n}A_0 = \{a_1, \ldots, a_n\}.

Taking ϕ(y,w)\phi(y, w) to be the equality predicate y=wy = w, we see Aacl(A)A \subseteq acl(A).

For idempotence of aclacl, suppose b 1,,b kacl(A)b_1, \ldots, b_k \in acl(A) and Mψ(c,b 1,,b k)\mathbf{M} \models \psi(c, b_1, \ldots, b_k) where {yM:Mψ(y,b 1,,b k)}\{y \in M: \mathbf{M} \models \psi(y, b_1, \ldots, b_k)\} has exactly mm elements. Write down a formula F m,ψ(x 1,,x k)F_{m, \psi}(x_1, \ldots, x_k) that says {yM:Mψ(y,x 1,,x k)}\{y \in M: \mathbf{M} \models \psi(y, x_1, \ldots, x_k)\} has at most mm elements (by adding some extra inequalities, we can make this “exactly mm elements”):

F m,ψ(x 1,,x k) y 1,,y m yψ(y,x 1,,x k)(y=y 1)(y=y m).F_{m, \psi}(x_1, \ldots, x_k) \coloneqq \exists_{y_1, \ldots, y_m} \forall_y \psi(y, x_1, \ldots, x_k) \Leftrightarrow (y = y_1) \vee \ldots \vee (y = y_m).

Now for i=1,,ki = 1, \ldots, k, let ϕ i\phi_i be a formula witnessing b iacl(A)b_i \in acl(A), i.e., Mϕ i(b i,a i,1,,a i,n i)\mathbf{M} \models \phi_i(b_i, a_{i, 1}, \ldots, a_{i, n_i}) where a i,kAa_{i, k} \in A and {xM:Mϕ i(x,a i,1,,a i,n i)}\{x \in M: \mathbf{M} \models \phi_i(x, a_{i, 1}, \ldots, a_{i, n_i})\} has finitely many elements. Then

x 1,,x kF m,ψ(x 1,,x k)ψ(y,x 1,,x k) i=1 kϕ i(x i,a i,1,,a i,n i)\exists_{x_1, \ldots, x_k} F_{m, \psi}(x_1, \ldots, x_k) \wedge \psi(y, x_1, \ldots, x_k) \wedge \bigwedge_{i=1}^k \phi_i(x_i, a_{i, 1}, \ldots, a_{i, n_i})

is a formula with parameters in AA that witnesses cacl(A)c \in acl(A). This proves the idempotence of aclacl.

As before, this notion of algebraic closure acl:P(M)P(M)acl: P(M) \to P(M) can be restricted to a closure operator acl X:P(X)P(X)acl^X: P(X) \to P(X) for a subset XMX \subseteq M, via the definition acl X(A)Xacl(A)acl^X(A) \coloneqq X \cap acl(A) for AXA \subseteq X. One often writes just aclacl instead of acl Xacl^X, provided that XX is understood.

Definitions

For an L\mathbf{L}-structure M\mathbf{M}, let DM nD \subseteq M^n be definable. DD is minimal if the only definable subsets of DD are finite or cofinite in DD. Slightly abusing language, if ϕ(x 1,,x n,a 1,,a k)\phi(x_1, \ldots, x_n, a_1, \ldots, a_k) is a formula with parameters that defines DD, we also say ϕ\phi is minimal. We say DD (or ϕ\phi) is strongly minimal if it is minimal in any elementary extension of M\mathbf{M}.

A theory T\mathbf{T} is strongly minimal if for any model M\mathbf{M} of T\mathbf{T}, the underlying set MM (definable by the formula x=xx = x) is strongly minimal.

Examples
  1. Let KK be an algebraically closed field. (Elimination of quantifiers, Chevalley’s theorem, etc.) Conclusion: ACF is a strongly minimal theory.

  2. Divisible torsionfree abelian groups.

  3. Non-example of dense linear orders.

Theorem

(Baldwin, Lachlan) The algebraic closure operator on a minimal set XX is a pregeometry.

Proof

Let AXA \subseteq X and suppose cacl(A{b})acl(A)c \in acl(A \cup \{b\}) - acl(A) for b,cXb, c \in X; we want to show bacl(A{c})b \in acl(A \cup \{c\}). Thus, suppose ϕ(c,b)\phi(c, b) is a formula with parameters from AA such that Mϕ(c,b)\mathbf{M} \models \phi(c, b) and card({xX:Mϕ(x,b)})=ncard(\{x \in X: \mathbf{M} \models \phi(x, b)\}) = n, a finite number. As in the proof of Proposition 1, let ψ(w)\psi(w) be a formula that says card({xX:Mϕ(x,w)})=ncard(\{x \in X: \mathbf{M} \models \phi(x, w)\}) = n. This is a formula with parameters from AA and ψ(b)\psi(b) is satisfied in M\mathbf{M}.

If {yX:Mϕ(c,y)ψ(y)}\{y \in X: \mathbf{M} \models \phi(c, y) \wedge \psi(y)\} is finite, then since bb belongs to this set, ϕ(c,y)ψ(y)\phi(c, y) \wedge \psi(y) would witness bacl(A{c})b \in acl(A \cup \{c\}) and we would be done. So suppose otherwise. Then this set is cofinite in XX, so that

card(X{yX:Mϕ(c,y)ψ(y)})=mcard(X - \{y \in X: \mathbf{M} \models \phi(c, y) \wedge \psi(y)\}) = m

for some finite mm, and we can again write down a formula χ(x)\chi(x) with parameters from AA that says

card(X{yX:Mϕ(x,y)ψ(y)})=m.card(X - \{y \in X: \mathbf{M} \models \phi(x, y) \wedge \psi(y)\}) = m.

If χ(x)\chi(x) defines a finite subset of XX, then since χ(c)\chi(c) is satisfied, we would reach the conclusion that cacl(A)c \in acl(A), a contradiction. Hence χ(x)\chi(x) defines a subset cofinite in XX. We may therefore choose n+1n+1 elements a 1,,a n+1Xa_1, \ldots, a_{n+1} \in X such that χ(a i)\chi(a_i) is satisfied. By our supposition,

B i{uX:Mϕ(a i,u)ψ(u)}B_i \coloneqq \{u \in X: \mathbf{M} \models \phi(a_i, u) \wedge \psi(u)\}

is cofinite for i=1,,n+1i = 1, \ldots, n+1. Hence i=1 n+1B i\bigcap_{i=1}^{n+1} B_i is inhabited, say by an element bb'. We have at least n+1n+1 elements xXx \in X such that ϕ(x,b)\phi(x, b') is satisfied, namely x=a 1,,a n+1x = a_1, \ldots, a_{n+1}. But now this contradicts the fact ψ(b)\psi(b').

References

  • Anand Pillay, Geometric stability theory, Oxford Logic Guides 32

  • slides from conference ”Geometric model theory”, Oxford 2010: directory html

  • Misha Gavrilovich, Model theory of universal covering space of complex algebraic varieties, thesis, pdf

  • Boris Zilber, Elements of Geometric Stability Theory, 2003 (pdf)


  1. Although oddly enough, as explained at the MacTutor biography page, what is called the Steinitz exchange condition was set out by Ernst Steinitz in a 1913 publication on convergent series and apparently not (?), as might be supposed, in his Algebraische Theorie der Körper (Crelle’s Journal, 1910). His 1913 lemma was for vector spaces.

Revised on February 17, 2014 15:59:34 by Todd Trimble (67.81.95.215)