model-theoretic Galois theory




Classical Galois theory is about the classification of intermediate field extensions in a (field-theoretic) algebraic closure of a ground field according to the structure of a profinite automorphism group.

This can be adapted to the setting of a universal domain of any first-order theory which eliminates imaginaries: we can analogously classify definably closed subsets of a model-theoretic algebraic closure of some parameter set according to the structure of a profinite automorphism group.

In the language of Grothendieck's Galois theory, we can sketch the next two sections as: the category of finite AA-definable sets in a monster model 𝕄\mathbb{M} equipped with the forgetful functor to Set is a Galois category.

The case of finite extensions

It will be instructive to look at the case of finite extensions first, as the case of infinite extensions will be a version of the same argument but souped-up with formalities about profinite groups.

Recall the fundamental theorem of Galois theory for finite extensions of fields:

Theorem. For L/KL/K a finite, separable, normal field extension, there is an order-reversing bijective correspondence

Sub Grp(Aut(L/K))Int(L/K)\operatorname{Sub}_{\mathbf{Grp}} \left( \operatorname{Aut}(L/K)\right) \leftrightarrows \operatorname{Int} \left(L/K\right)

between the subgroups of the group of field automorphisms of LL fixing KK pointwise, and the intermediate field extensions between KK and LL. The correspondence is given by sending a subgroup to its field of fixed points, and an intermediate extension to its stabilizer subgroup.

This can be translated to general model-theoretic language as follows: we work in a monster model 𝕄T\mathbb{M} \models T for a complete first-order theory which eliminates imaginaries. Let AA be a small parameter set in MM. We have the following

Dictionary, between our general TT eliminating imaginaries and the special case T=[[ACF]]T = [[ACF]] the theory of an algebraically closed field.

  • AKA \supseteq K corresponds to “AA is an extension of KK”.
  • If AA is an extension of KK, the definable closure of AA corresponds to the perfect hull of the field K(A)K(A) generated by adjoining AA.
  • If AA is an extension of KK, the algebraic closure of AA corresponds to the separable closure of the field K(A)K(A) generated by adjoining AA.
  • If AA is a definably closed extension of KK with A=dcl(γK)A = \operatorname{dcl}(\gamma \cup K) for γ\gamma finite, this corresponds to K(A)K(A) being finitely-generated.
  • The orbit of an element 𝕄\ell \in \mathbb{M} under Aut(𝕄/K)\Aut(\mathbb{M}/K) corresponds to conjugates of \ell in K()K(\ell) under Aut(K()/K)\operatorname{Aut}(K(\ell)/K).
  • The size of this orbit corresponds to the degree of the field extensions K()/KK(\ell)/K.
  • If this orbit is finite, \ell is said to be algebraic over KK.
  • The condition “LK\forall \ell \in L \supseteq K, Orb Aut(𝕄/L)()L\operatorname{Orb}_{\operatorname{Aut}(\mathbb{M}/L)}(\ell) \subseteq L” corresponds to “LL is a normal extension of KK”.

When LL is a normal extension of KK, LL splits into Aut(𝕄/K)\operatorname{Aut}(\mathbb{M}/K)-orbits, and so the latter group acts via restriction on L/KL/K. We call the image of the induced group homomorphism Aut(𝕄/K)Sym(L/K)\operatorname{Aut}(\mathbb{M}/K) \to \Sym(L/K) Aut(L/K)\operatorname{Aut}(L/K).

With this in place, we can show:

Theorem. Let KK be a definably closed parameter set. Let AA be a normal extension of KK generated by the finite algebraic tuple γ\gamma. Then there is an order-reversing bijective correspondence between the subgroups of Aut(A/K)\operatorname{Aut}(A/K) and the definably closed intermediate extensions of A/KA/K. The correspondence is given by maps Fix\mathsf{Fix} sending a subgroup to its fixed points and Stab\mathsf{Stab} sending an intermediate definably closed extension to its stabilizer subgroup.

Proof. By saturation, Fix\mathsf{Fix} is well-defined, and Stab\mathsf{Stab} is clearly well-defined.

Fix\mathsf{Fix} is left-inverse to Stab\mathsf{Stab}: by saturation in the monster, any fixed points of Stab(B)\mathsf{Stab}(B) for KBAK \subseteq B \subseteq A must be in the definable closure of BB, so whenever BB is definably closed, Fix(Stab(B))B\mathsf{Fix} \left( \mathsf{Stab} (B) \right) \subseteq B, with the reverse inclusion immediate.

Stab\mathsf{Stab} is left-inverse to Fix\mathsf{Fix}: for HH a subgroup of Aut(A/K)\operatorname{Aut}(A/K), note that for any cc a code for the HH-orbit of γ\gamma and for any σAut(𝕄/K)\sigma \in \operatorname{Aut}(\mathbb{M}/K), σ\sigma fixes cc if and only if the restriction σrestrictionA\sigma \restriction A permutes H.γH.\gamma. Since γ\gamma generates AA, any automorphism in Aut(A/K)\operatorname{Aut}(A/K) is determined by where it sends γ\gamma, so σrestrictionA.γH.γσrestrictionAH\sigma \restriction A \operatorname{.} \gamma \in H.\gamma \iff \sigma \restriction A \in H.

In particular, since H.γH.\gamma is finite, cc is actually H.γH.\gamma-definable, hence AA-definable. Since AA is definably closed, cAc \in A, and so cFix(H)c \in \mathsf{Fix}(H). If gStab(Fix(H))g \in \mathsf{Stab}(\mathsf{Fix}(H)), gg in particular fixes cc, hence gHg \in H, and it’s clear that HStab(Fix(H))H \subseteq \mathsf{Stab}(\mathsf{Fix}(H)). \square

The case for infinite extensions

We’ll get the case for infinite extensions by just classifying all subextensions of acl(A)/dcl(A)\operatorname{acl}(A)/\operatorname{dcl}(A) at once.

The model-theoretic absolute Galois group

Let 𝕄T\mathbb{M} \models T be a monster model. Let AA be a small parameter set. acl(A)\operatorname{acl}(A) is a normal extension of AA, because every finite AA-definable set splits into Aut(𝕄/A)\operatorname{Aut}(\mathbb{M}/A)-orbits. The absolute Galois group Gal(A)\operatorname{Gal}(A) of AA is Aut(acl(A)/dcl(A)\operatorname{Aut}(\operatorname{acl}(A)/\operatorname{dcl}(A).

(For example, in ACF\mathsf{ACF}, this recovers the usual absolute Galois group.)

Now, acl(A)\operatorname{acl}(A) is the colimit of the diagram of finite AA-definable sets. From commutativity of limits and colimits, we know that whenever F:CG-SetF : \mathbf{C} \to G \text{-} \mathbf{Set} is a cofiltered diagram of GG-sets, then taking orbits of limF\underset{\longleftarrow}{\lim}F is the same as taking a limit of the orbits. Dually, if we take automorphism groups, we get:

Aut(limF)lim(F(c)). \operatorname{Aut}\left(\underset{\longrightarrow}{\lim} F\right) \simeq \underset{\longleftarrow}{\lim} \left(F(c) \right).

So Gal(A)\operatorname{Gal}(A) is profinite.

The fundamental theorem of model-theoretic Galois theory

Theorem. Let TT be a first-order theory which eliminates imaginaries, and let 𝕄T\mathbb{M} \models T be a monster. Let A𝕄A \subseteq \mathbb{M} be a small parameter set. Then there is a bijective order-reversing correspondence

Fix:Sub Krull-closed(Gal(A))Sub dcl-closed(acl(A)/dcl(A)):Stab \mathsf{Fix} : \operatorname{Sub}_{\text{Krull-closed}}\left(\operatorname{Gal}(A)\right) \leftrightarrows \operatorname{Sub}_{\text{dcl-closed}}\left(\operatorname{acl}(A)/\operatorname{dcl}(A)\right) : \mathsf{Stab}

given by taking a subgroup closed in the profinite topology of Gal(A)\operatorname{Gal}(A) to its fixed points and by taking a definably-closed intermediate extension of acl(A)/dcl(A)\operatorname{acl}(A)/\operatorname{dcl}(A) to its stabilizer.

Proof. That Fix\mathsf{Fix} is left-inverse to Stab\mathsf{Stab} again follows from being in a monster.

On the other hand, let HH be a closed subgroup. HH is an intersection of basic open subgroups H iH_i which are preimages of Aut(B i/A)\operatorname{Aut}(B_i/A), for B iB_i finite and AA-definable. Fixing an ordering on the B iB_i and treating them as tuples, obtain codes c ic_i for the orbit of each B iB_i under Aut(B i/A)\operatorname{Aut}(B_i/A). Since each B iB_i is finite AdefinableA-definable, c ic_i is AA-algebraic. The stabilizer of each c ic_i is precisely H iH_i, so H= iIStab(c i)H = \bigcap_{i \in I} \mathsf{Stab}(c_i). Since each c ic_i is fixed by HH, whenever gStab(Fix(H))g \in \mathsf{Stab} \left(\mathsf{Fix}(H) \right), then in fact gHg \in H. \square

Model theory and the Tannakian formalism

In the motivating examples (see below) it turns out that Galois (i.e. relative automorphism) groups are themselves definable (i.e. arise as interpretations in the model of internal groups in Def(T)\mathbf{Def}(T).) In this case what you’re taking the Galois group of must formally resemble an internal diagram on this internal group; in model theory these are studied as what model theorists call internal covers. A structure theorem of Hrushovski makes this correspondence explicit: internal covers are torsors of definable groupoids and vice-versa; see here.

All internal groups in Def(ACF)\mathbf{Def}(\mathsf{ACF}) are in fact algebraic groups, so this is reminiscent of reconstruction results arising from Tannaka duality. As it turns out, one can make this analogy explicit and recover a slightly-weakened version of the Tannakian formalism for algebraic groups using the theory of internal covers; see the paper by Moshe Kamensky here.


If T=ACFT = \mathsf{ACF} the theory of algebraically closed fields, this recovers the classical fundamental theorem of Galois theory.

If T=DCFT = \mathsf{DCF} the theory of differentially closed fields?, this recovers differential Galois theory. (In fact, Kolchin’s work was what inspired Poizat to introduce imaginaries and work out classical Galois theory in a model-theoretic setting.)

Any theory TT can be conservatively interpreted inside a theory T eqT^{\operatorname{eq}} which eliminates imaginaries and hence “admits a Galois theory.” This is the coherent special case of a result Olivia Caramello spells out in very general terms in her article (Caramello 13) on topological (toposic) Galois theory.



Last revised on May 6, 2020 at 06:01:33. See the history of this page for a list of all contributions to it.