nLab model-theoretic Galois theory

Idea

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 $A$-definable sets in a monster model $\mathbb{M}$ 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 $L/K$ a finite, separable, normal field extension, there is an order-reversing bijective correspondence

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

between the subgroups of the group of field automorphisms of $L$ fixing $K$ pointwise, and the intermediate field extensions between $K$ and $L$. 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 $\mathbb{M} \models T$ for a complete first-order theory which eliminates imaginaries. Let $A$ be a small parameter set in $M$. We have the following

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

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

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

With this in place, we can show:

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

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

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

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

In particular, since $H.\gamma$ is finite, $c$ is actually $H.\gamma$-definable, hence $A$-definable. Since $A$ is definably closed, $c \in A$, and so $c \in \mathsf{Fix}(H)$. If $g \in \mathsf{Stab}(\mathsf{Fix}(H))$, $g$ in particular fixes $c$, hence $g \in H$, and it’s clear that $H \subseteq \mathsf{Stab}(\mathsf{Fix}(H))$. $\square$

The case for infinite extensions

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

The model-theoretic absolute Galois group

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

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

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

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

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

The fundamental theorem of model-theoretic Galois theory

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

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

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

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

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

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 $\mathbf{Def}(T)$.) 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 $\mathbf{Def}(\mathsf{ACF})$ 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.

Examples

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

If $T = \mathsf{DCF}$ 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 $T$ can be conservatively interpreted inside a theory $T^{\operatorname{eq}}$ 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.

Discussion

Related concepts

• Galois theory
• Grothendieck's Galois theory
• elimination of imaginaries
• algebraic type
• Tannaka duality

References

• Bruno Poizat, Une Theorie de Galois Imaginaire.

• Bruno Poizat, A Course in Model Theory.

• Alice Medvedev and Ramin Takloo-Bighash, An invitation to model-theoretic Galois theory

• Olivia Caramello, Topological Galois theories, (arXiv: 1301.0300)