Understanding Stone Duality: Boolean Rings to Stone Spaces

1. Stone Duality (forward direction) By Eric Yu, Class of 2026

2. Abstract The contravariant functor from Boolean rings to topological spaces sending A to Spec(A) is fully faithful, and therefore a dual equivalence onto its essential image, which turns out to be the category of Stone spaces (i.e. topological spaces which are compact, Hausdorff, and totally disconnected). In this talk, we will derive some properties of boolean rings and see how those properties translate through the Zariski topology to the properties of Stone spaces.

3. Definition: Boolean Ring A boolean ring R is a ring with the property that every element equals its square: ∀ x ∈ R, x2= x. Example: (ℤ/2ℤ)n

4. Properties of Boolean Rings 1 + 1 = 0. Corollary: 2x = 0 for all x in R.

5. Properties of Boolean Rings ℤ/2ℤ is the only integral domain boolean ring.

6. Properties of Boolean Rings The quotient of a boolean ring is a boolean ring. Proof: Just apply the quotient map to both sides of x2= x.

7. Properties of Boolean Rings Every prime ideal is maximal with index 2.

8. More Examples of Boolean Rings Our example (ℤ/2ℤ)ncan be generalized. The power set of any set S, with symmetric difference as addition and intersection as multiplication, forms a boolean ring.

9. Even More Examples of Boolean Rings We can get even more general: Instead of taking all of P(S), we just need to take a subset of P(S) which contains ∅ and S, and is closed under intersection and symmetric difference. Example: subsets of ℤ which are either finite or have finite complement. Incredible fact: Every boolean ring can be constructed in this way.

10. One More Property of Boolean Rings Notice that A + B + AB = A ∪ B. This motivates our next property:

11. ⟨a, b⟩ denotes the ideal generated by a and b. One More Property of Boolean Rings ⟨a, b⟩ = ⟨a + b + ab⟩. Corollary: Every finitely generated ideal is principal.

12. Stone Spaces A Stone space is a topological space that is compact, Hausdorff, and totally disconnected.

13. Examples of Stone Spaces ● The only finite Hausdorff spaces have the discrete topology, so all finite Stone spaces are discrete. ● The subspace of the reals {0, 1, ½, ⅓, ¼, … } is a Stone space. ● The Cantor set is a Stone space.

14. Spec(A) Given a commutative ring A, let X be the set of prime ideals of A. Let P denote the power set. Define the function V: P(A) → P(X) by having it send any set of elements in A to the set of prime ideals which contains all of those elements. We define Spec(A) to be a topological space whose underlying set is X and whose closed sets are im(V). This topology is called the Zariski topology. V(E) = V(⟨E⟩), so we can restrict the domain of V to the set of ideals of A.

15. Spec(A) We will soon see that if A is a boolean ring, then Spec(A) is a Stone space. First, let’s deduce some general facts about Spec(A).

16. Is Spec(A) Really a Topological Space? We must show that im(V) is closed under arbitrary intersection and finite union, and that it includes the empty set and all of X.

17. Example: Spec(ℤ) X = { ⟨0⟩, ⟨2⟩, ⟨3⟩, ⟨5⟩ . . . }. Since ℤ is a PID, the set of ideals of ℤ is { ⟨0⟩, ⟨1⟩, ⟨2⟩, ⟨3⟩ . . . }. V(⟨n⟩) = { ⟨p⟩ | p is in the prime factorization of n }. All V(⟨n⟩) are finite, and a finite set {⟨a⟩, ⟨b⟩, ⟨c⟩} ⊆ X ∖ {⟨0⟩} is equal to V(⟨abc⟩). So the closed sets in Spec(ℤ) are precisely those which are finite and do not contain ⟨0⟩. (or all of X.) Conclusion: Spec(ℤ) is the cofinite topology on the set of all prime numbers, with the generic point ⟨0⟩ attached.

18. Theorem: Spec(A) is Compact Given a ∈ A, we define D(a) to equal the open set X ∖ V({a}), the set of all prime ideals which do not contain a. Fact: The D(a)’s form a basis of open sets. Given any open cover of Spec(A), it can be refined so that every set in the cover is of the form D(a) for some a in A. We want to show that this has a finite subcover.

19. Theorem: Spec(A) is Compact Lemma: The D(ai)’s cover X if and only if the ai’s generate A.

20. Theorem: Spec(A) is Compact So we want to show that a finite subset of the ai’s generate the entire ring. This is easy: we can express 1 in terms of a finite number of the ai’s, and those ai’s generate the entire ring (since in particular they generate 1). And so we’ve done it.

21. Theorem: If A is a Boolean Ring, then Spec(A) has a Clopen Basis We will show that for all a ∈ A, we have D(a) = V({a + 1}), meaning D(a) is clopen. This statement is equivalent to saying a prime ideal does not contain a if and only if it contains a + 1. Proof: A prime ideal cannot contain both a and a + 1 or else it would contain 1. A prime ideal cannot contain neither a nor a + 1 or else it would not contain a(a + 1) = a2+ a = a + a = 2a = 0.

22. Theorem: If A is a Boolean Ring, then Spec(A) is Hausdorff and Totally Disconnected. We will show that given distinct prime ideals x and y in X, there exists a partition of X into two disjoint open sets, each containing one of x and y. Without loss of generality, assume x ⊈ y. Then there exists some a ∈ x ∖ y. Then D(a) is a clopen set containing y but not x, and we’re done. Another way to see this: Compact + Hausdorff + Clopen basis ⟹ Totally disconnected

23. Examples to Think About: In this talk, we have seen that Spec maps commutative rings to topological spaces, and that in particular it maps boolean rings to Stone spaces. Here are some examples: ● (ℤ/2ℤ)nis mapped to the discrete topology on n points. ● The subsets of ℤ which are either finite or have finite complement (with intersection and symmetric difference as multiplication and addition) is mapped to {0, 1, ½, ⅓, ¼, … } ⊆ ℝ. ● ?2[x1, x2, … ] / ((x12– x1), (x22– x2), … ) is mapped to the Cantor set.

24. Facts Left as an Exercise: ● Spec is a contravariant functor from the category of commutative rings to the category of topological spaces. That is, it respects and reverses morphisms. ● Given a Stone space, we can recover the boolean ring that maps to it by taking the set of clopen sets with the operations of intersection and symmetric difference.

25. Thank You! Special thanks to Marc.