Spec R, The Spectrum of a Boolean Ring

The Spectrum of a Boolean Ring

Let R be a boolean ring. Let e be an element of R, hence e is idempotent. Since e and 1-e span R, v_e and v_1-e have no primes in common. In other words, the intersection of these two base closed sets is empty. At the same time, their union is v_e*(1-e) = v₀ = everything. Each base closed set v_e is open, and each base open set o_e is closed.

We would like to show the converse, that the clopen sets are base sets in the zariski topology.

Let e₁ through e_n generate base open sets. The complement of their union is v_H, where H is the ideal generated by e₁ through e_n. Every finitely generated ideal is principal, so let f generate H. The union of our base sets is o_f, another base set.

If y is open and closed, then it is the union of base sets, and being a closed subset of a compact space, it is the finite union of base sets. Therefore y is a base set, and the open and closed sets are exactly the base sets.

Spec R is Hausdorff

Let G and H be prime ideals, such that H does not contain G. Let x lie in G-H. Now v_x and o_x are disjoint open sets that separate G and H.

Stone's Theorem

Stone's theorem states that any boolean lattice is the collection of clopen sets in some compact hausdorff topological space, with meet being intersection and join being union.

Convert the boolean lattice to a ring, and then derive spec R. The clopen sets in spec R correspond to the elements of R, which are the elements of the lattice. For x and y in R, the intersection of o_x and o_y is o_xy, and their union is o_x+y+xy, since x+y+xy is the element that generates the principal ideal spanned by x and y. These are the meet and join respectively. The lattice has been mapped onto clopen sets, with join as union and meet as intersection.

Is the map injective? Given distinct points in the lattice, assume x is not below y. If the ideal R*y contains x then x = zy, and x is below y in the lattice, which is a contradiction. The ideal generated by y misses x, and the powers of x. Drive this up to a prime ideal containing y, and missing x. This shows v_y ≠ v_x, and clopen sets faithfully represent the lattice.