Remember this challenge? Probably not. But wait, don’t click! Namely, I will do something more general in this post.
Suppose we have a ring A and a contravariant functor F on (Sch/A) with the following properties:
- F satisfies the sheaf property for fpqc coverings
- the value of F on a scheme is either a singleton or empty
- for every quasi-compact scheme T/A such that F(T) is nonempty, there is an ideal I of A such that F(Spec(A/I)) is nonempty and such that T —> Spec(A) factors through Spec(A/I).
Example: A = k[x, y] for a field k and F(T) is nonempty if and only if the generic point of Spec(A) is not in the image of T —> Spec(A). Here F is not a representable functor.
I’d like to add some conditions that guarantee that F is representable by a closed subscheme of Spec(A). Here is what I just came up with; I think it is obviously correct and the right thing to do. If A is Noetherian we add the following two conditions
- If s_1, s_2, s_3, … is an infinite sequence of points of Spec(A) such that F(s_i) is nonempty and s is a limit point of the sequence, then F(s) is nonempty.
- If A —> …. —> A_n —> A_{n – 1} —> … —> A_1 are surjections such that the kernels A_n —> A_{n – 1} are locally nilpotent ideals and F(Spec(A_i)) is nonempty, then F(Spec(A_∞)) is nonempty where A_∞ = lim A_n.
I leave it as an exercise to show that 1 — 5 imply F is representable in the desired manner. If A is not Noetherian, then somehow these should still be enough although maybe you need to replace the natural numbers by a bigger directed set.
Why am I excited by this observation? It is because I want to apply this to the situation of the other blog post mentioned above: X is an algebraic space of finite presentation over A, u : H —> G is a map between quasi-coherent O_X-modules. We assume G is flat over A, of finite presentation, and universally pure relative to A (this is a technical condition which is satisfied if the support of G is proper over A). The functor F is defined by F(T) is nonempty if and only if the base change u_T of u is zero.
Properties 1, 2, 3 hold for F and are easy to prove. The proof of property 4 still doesn’t use purity of G relative to A (I think because we already have 3 it follows from an argument using generic freeness, but I also have an argument using \’etale localization). The key is to prove property 5.
To see 5 is true, I argue as follows. Suppose that the base change u_∞ to A_∞ is nonzero. Choose a weakly associated point ξ of the image of u_∞. This is also a weakly associated point of G_∞. The image t’ of ξ in Spec(A_∞) specializes to a point t in V(I_1) = Spec(A_1) because I_1 is contained in the radical of A_∞. Because G is universally pure relative to A, there is a specialization θ of ξ which lies over t (indeed this is the definition of being pure relative to the base). Then since u_∞ is zero at θ (in a suitable \’etale neighbourhood Edit: Argh… I just discovered this doesn’t work!) it is zero at ξ, a contradiction.
Enjoy!
PS: A finitely presented module G on X flat and pure over A is universally pure relative to A. However, this is harder to prove than the above and it is easy to see that support proper over A implies universal purity.
