Recall that the petit étale topos of a scheme X is the topos of sheaves on the category of schemes étale over X (with respect to the étale topology). Since étale morphisms are open, any closed sieve on X must contain the union of the (open) images of all its members. Thus, the closed sieves on X are classified by the open subspaces of X, so the localic reflection of the petit étale topos of X is the petit Zariski topos of X.
Note also that the same argument works for the petit fppf topos of X – the point being that flat morphisms locally of finite presentation are open.
By applying the functional approach to general topology proposed by Clementino, Giuli, and Tholen, we may make a formal analogy between flat morphisms of affine schemes and open maps of topological spaces. First, the set-up:
- We are given a category 𝒞 and a class ℱ of morphisms in 𝒞; for convenience, we assume 𝒞 has pullbacks.
- A proper morphism is a morphism f : X → Y such that, for any pullback f′ : X′ → Y′ of f : X → Y along any morphism Y′ → Y, f′ : X′ → Y′ is a member of ℱ.
- A closed embedding is a monomorphism in 𝒞 that is also a proper morphism.
- A dominant morphism is a morphism that is left orthogonal to every closed embedding.
- A open morphism is a morphism f : X → Y such that pullback along f preserves dominant morphisms in the following sense: for any pullback f′ : X′ → Y′ of f : X → Y along any morphism Y′ → Y, the pullback of any dominant morphism Y″ → Y′ along f′ : X′ → Y′ is a dominant morphism X″ → X′.
If we take 𝒞 to be the category of topological spaces and ℱ to be the class of closed maps (i.e. continuous maps such that the image of every closed subset is again a closed subset), then these definitions agree with the classical ones. On the other hand, we could take 𝒞 to be the category of affine schemes and ℱ to be the class of closed immersions, in which case the dominant morphisms correspond to injective homomorphisms of commutative rings.
Unfortunately, the analogy ends there: although open immersions of schemes are flat monomorphisms, they are also required to be locally of finite presentation, which is not automatic. There are also some subtleties when trying to apply these definitions to the category of all schemes: although a dominant morphism in the sense above is a morphism whose scheme-theoretic image is the whole codomain, the connection between open morphisms in the sense above and flat morphisms of schemes is much less obvious.
It had long puzzled me why people usually use the ideal product to show that the intersection of any two Zariski-open subsets of the prime spectrum of a ring is again a Zariski-open subset – after all, the intersection of the two ideals would do just as well. But perhaps this is the reason:
Proposition. Let f : A → B be a ring homomorphism and let I and J be ideals of A. Then the ideal of B generated by f(IJ) is the ideal product of the ideals generated by fI and fJ.
It is well known that the poset of ideals of a ring has arbitrary joins (namely, the ideal sum) and that they are preserved by pushforward along ring homomorphisms. Moreover, the ideal product distributes over the ideal sum, so the poset of ideals of a ring in fact a quantale, and we have a functor CRing → Quant. A result of Mulvey says that the frame of Zariski-open subsets of the prime spectrum of A is the frame reflection of the quantale of ideals of A.
While it is true that the poset of ideals is a complete lattice, not all binary meets (i.e. intersections) are preserved by pushforward. (Consider the homomorphism k[x, y] / (xy) → k[z] / (z2) sending x and y to z.) Distributivity also fails. (Consider the ring k[x, y] / (x2, xy, y).) Still, one might ask, what is the relationship between the frame of Zariski-open subsets of the prime spectrum and the frame reflection of the poset of ideals, considered as a join semilattice with finite meets?
Let X be a scheme. The following are equivalent:
- X is a local scheme, i.e. of the form Spec A where A is a local ring.
- X is a scheme with a unique closed point x and the only (open) neighbourhood of x is X itself.
- Every open covering of X must contain X itself.
It is well known that the first condition implies the second, and it is an easy exercise (in point set topology) to show that the second condition implies the third. To complete the proof of the claim, it is enough to show that the third condition implies the first.
Suppose X is a scheme satisfying the third condition. Any scheme can be covered by open affine subschemes, so the condition on X implies it is affine. In particular, it has a closed point x. The condition also implies that the complement of x is the unique maximal open proper subset of X, so x must be the unique closed point of X, as required.
We should observe that the third condition can be expressed purely in terms of the Zariski topology on the category of schemes: it says that a scheme X is local if and only if every Zariski-covering family of X contains a split epimorphism, or equivalently, if and only if there is a unique Zariski-covering sieve on X (namely, the maximal sieve).