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.
Let 𝒞 be a category and let J be a subcanonical Grothendieck topology on 𝒞. A J-local object in 𝒞 is an object X in 𝒞 such that the only J-covering sieve on X is the maximal sieve. For example, if 𝒞op is the category of commutative rings and J is the Zariski topology, then the J-local objects are precisely the local rings; or if 𝒞 is the category of T₁-spaces and J is the open cover topology, then J-local objects are precisely the points.
It is easy to see that representable J-sheaves on 𝒞 corresponding to J-local objects in 𝒞 are projective (with respect to sheaf epimorphisms). However, the converse is not true in general; after all, the class of projective objects is closed under coproduct, whereas the class of J-local objects in 𝒞 may fail to be closed under coproduct. For example, if 𝒞 is the category of (all) topological spaces and J is the open cover topology, then any representable sheaf corresponding to a discrete space is projective. In fact, a representable sheaf is projective if and only if every open cover of the corresponding topological space can be refined by a cover of clopen subsets.
So, whereas J-local objects in 𝒞 should be thought of as being point-like, it would seem that objects in 𝒞 such that the corresponding representable sheaf is projective should be thought of as being “highly atomised”, but not necessarily a coproduct of J-local objects. This is one heuristic explanation for why we should not expect sheaf toposes to have enough projectives.
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).