Galois correspondences

David Pierce

2004.12.16

This note gives a unified treatment to several mathematical observations.

For notation, let 2 be considered as the universe {0, 1} of an abelian group. For each e in 2, let A_{e} be a set, and let ∗ be a relation from A_{0} to A_{1}. This means that ∗ is a subset of A_{0} × A_{1}; if ∗ contains (x_{0}, x_{1}), then we write

x_{0} ∗ x_{1}.

If x_{e} ∈ A_{e}, let

[x_{e}] = {x_{e+1} ∈ A_{e+1} : x_{0} ∗ x_{1}} ∈ 𝒫(A_{e+1}).

If X ∈ 𝒫(A_{e}), let

X′ = ⋂{[x] : x ∈ X} ∈ 𝒫(A_{e+1}).

Let A_{e+1}^{∗} be the image of 𝒫(A_{e}) under the map X ↦ X′.

Theorem 1. Let e ∈ 2.

(∗) If X, Y ∈ 𝒫(A_{e}) and X ⊆ Y, then Y′ ⊆ X′.
(†) If X ∈ 𝒫(A_{e}), then X ⊆ X′′.

(‡) The map X ↦ X′ is a bijection from A_{e}^{∗} to A_{e+1}^{∗} with inverse X ↦ X′.

Proof. Exercise. (For the last point, see [3, ch. V, Lemma 2.6].)

Regardless of how the maps X ↦ X′ are originally defined, if they meet the conditions established in the theorem, they constitute a Galois correspondence between A_{0}^{∗} and A_{1}^{∗}. (This definition is in [4, p. 35].) There are several examples, as you should verify:

Field-theory

The usual Galois correspondence in field-theory is the case when A_{0} is a field L that is a finite Galois extension of a field K, and A_{1} is Aut(L/K), and

x ∗ σ ⟺ x^{σ} = x.

Then A_{0}^{∗} comprises the subfields F of L that include K, and A_{1}^{∗} comprises the subgroups H of Aut(L/K), and F′ = Aut(L/F), and H′ = Fix(H).

The Zariski topology

Suppose A_{0} is a ring R (commutative with 1), and A_{1} is Spec R, that is, the set of prime ideals of R. Let ∗ be ∈. Then

[x] ∪ [y] = [xy]

if x, y ∈ R. Hence the sets [x] are the basic closed sets for a topology, the Zariski topology on Spec R. (See for example [1, pp. 54–55] or [2, § II.2].) The topology is compact, although possibly not Hausdorff. In this topology, if X ⊆ Spec R, then X′′ is the closure of X. If X ⊆ R, then X′′ is the radical (in the sense of [3, ch. VIII, Definition 2.5]) of the (possibly improper) ideal (X). In general, A_{0}^{∗} comprises the radical ideals of R, and A_{1}^{∗} comprises the closed subsets of Spec R.

The Stone space

Now suppose in particular that A_{0} is a Boolean ring or algebra B, and A_{1} is its Stone space S(B), the set of ultrafilters of B. The ultrafilters are dual to the prime ideals, all of which are maximal. Let ∗ be ∈ again. Then

[x] ∩ [y] = [xy] = [x ∧ y]

when x, y ∈ B, and also

[x]^{c} = [x + 1] = [¬x],

so that

[x] ∪ [y] = [x + y + xy] = [x ∨ y].

Hence the sets [x] are basic open and closed sets for a topology on S(B). This topology is compact as before, but also Hausdorff. The elements of A_{1}^{∗} are still just the closed subsets of S(B); the elements of A_{0}^{∗} are just the filters of B. If X ⊆ B, then X′′ is the filter generated by X; if X ⊆ S(B), then X′′ is its closure.

Model-theory

Suppose L is a signature for first-order logic. Let A_{0} be the class Mod(L) of L-structures, let A_{1} be Sn_{L}, and let ∗ be ⊧. Then A_{1}^{∗} is the set of theories of L, and A_{0}^{∗} is the set of elementary classes of L-structures. (See [4, § 3.4].)

References

[1] David Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.

[2] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

[3] Thomas W. Hungerford. Algebra, volume 73 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1980. Reprint of the 1974 original.

[4] Philipp Rothmaler. Introduction to model theory, volume 15 of Algebra, Logic and Applications. Gordon and Breach Science Publishers, Amsterdam, 2000. Prepared by Frank Reitmaier, Translated and revised from the 1995 German original by the author.