One point compactification

Let f : X —> S be a separated morphism of finite presentation. Consider the functor F : (Sch)^{opp} —> (Sets) which to a scheme T associates all pairs (a, Z) where a : T —> S and Z is a closed sub scheme of the base change X_T such that the projection Z —> T is an open immersion. In other words, this is the functor of flat families of closed sub schemes of degree <= 1 on X/S, as we discussed briefly in this post. As we saw there it is not true in general that F is an algebraic space. If X = A^1_S then F is (probably) a directed colimit of schemes. But if X has higher dimension I’m not sure how to “compute” F.

Here are some general properties of this construction. There is a canonical morphism

j : X —> F

which is an open immersion by construction. Moreover, there is a canonical morphism

∞ : S —> F

which associates to a : T —> S the pair (a, ∅). And of course on points we have F = j(X) ∪ ∞(S). The structure morphism p : F —> S is locally of finite presentation and satisfies the valuative criterion (both existence and uniqueness). These properties tell us p is “proper”. Thus F is morally speaking the one point compactification of X/S.

When I was discussing this with Bhargav Bhatt he suggested we think about the etale cohomology of F. Now that I have had some time to think about his suggestion, I think this is a splendid idea. Namely, it seems to me that we could try to use Rp_*Rj_! to define Rf_! for the morphism f : X —> S…

3 thoughts on “One point compactification

  1. The functor F is exactly what I use to define “The canonical embedding of an unramified morphism in an étale morphism” (see my paper with the same title). I only consider f:X->S unramified (but not necessarily separated) as otherwise F is not necessarily representable. Do note that F is NOT separated nor universally closed. If X->S is proper, then F is universally closed but F is almost never separated (it is separated if and only if X->S is étale and separated). For example, if X is a closed subscheme of S, then F(X/S) is the gluing of two copies of X along X-S.

    This is of course very closely related to étale cohomology as I did tell you half-a-year ago! In fact, the functor F can also be constructed as follows:

    Consider étale sheaves of pointed sets. For any morphism f:X->Y one can define an operation f_#: Shv(X)->Shv(Y) on sheaves of pointed sets. If f is unramified, then F(X/Y)=f_#{0,1}_X, if f is étale, then f_# is left adjoint to f^{-1} and if f is a monomorphism, then f_#=f_! is extension by zero. If I remember correctly, unless f is unramified (my definition of) f_# is rather boring.

    • My functor is different because if X —> S is a closed immersion then my F is representable by X \coprod S which is different from what you say. My functor always has existence and uniqueness of the valuative criterion as you can prove as follows: Take a valuation ring R with fraction field K and set T = Spec(R). Choose a morphism T —> S. Let X_T be the pullback. A point of F(K) corresponds either to ∞ or to a K-rational point x of X_K. In the first case the unique value of F(T) is ∞ again. In the second case you simply take Z \subset X_T be the scheme theoretic closure of x. This is a flat separated scheme of finite type over T whose generic fibre is Spec(K) and hence maps isomorphically to an open subscheme of T (I think — certainly OK for dvrs).

      My functor is almost never representable, and this is exactly what the post above was about. And what Bhargav was saying is that of course the cohomology of F still exists as we can localize the big etale site of at the sheaf F.

      • Sorry, got carried away. My functor is indeed different. I require that Z is an OPEN subscheme of X_T such that Z->T is a CLOSED immersion.

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